I have adapted the three solutions presented here (and modified one where it failed) to produce a single, simple setProxy method that works for all versions of Android. I've tested it from 10 to 18, and it works for all tested environments.

UPDATED 2014-04-04 I finally worked in the solution from a comment below, courtesy of nubela and xjy2061. Now this works for all current Android versions, including KitKat 4.4. If you implement your own Application subclass, supply the name of the class as the optional fourth parameter.

UPDATED 2015-01-15 The part labeled optional in the KitKat method throws an exception on Lollipop because an auxiliary class is missing, but without that it works on both KitKat and Lollipop, since the WebView is based on Chromium in both cases.

public static boolean setProxy(WebView webview, String host, int port, String applicationClassName="android.app.Application") {
    // 3.2 (HC) or lower
    if (Build.VERSION.SDK_INT <= 13) {
        return setProxyUpToHC(webview, host, port);
    }
    // ICS: 4.0
    else if (Build.VERSION.SDK_INT <= 15) {
        return setProxyICS(webview, host, port);
    }
    // 4.1-4.3 (JB)
    else if (Build.VERSION.SDK_INT <= 18) {
        return setProxyJB(webview, host, port);
    }
    // 4.4 (KK) & 5.0 (Lollipop)
    else {
        return setProxyKKPlus(webview, host, port, applicationClassName);
    }
}

/**
 * Set Proxy for Android 3.2 and below.
 */
@SuppressWarnings("all")
private static boolean setProxyUpToHC(WebView webview, String host, int port) {
    Log.d(LOG_TAG, "Setting proxy with <= 3.2 API.");

    HttpHost proxyServer = new HttpHost(host, port);
    // Getting network
    Class networkClass = null;
    Object network = null;
    try {
        networkClass = Class.forName("android.webkit.Network");
        if (networkClass == null) {
            Log.e(LOG_TAG, "failed to get class for android.webkit.Network");
            return false;
        }
        Method getInstanceMethod = networkClass.getMethod("getInstance", Context.class);
        if (getInstanceMethod == null) {
            Log.e(LOG_TAG, "failed to get getInstance method");
        }
        network = getInstanceMethod.invoke(networkClass, new Object[]{webview.getContext()});
    } catch (Exception ex) {
        Log.e(LOG_TAG, "error getting network: " + ex);
        return false;
    }
    if (network == null) {
        Log.e(LOG_TAG, "error getting network: network is null");
        return false;
    }
    Object requestQueue = null;
    try {
        Field requestQueueField = networkClass
                .getDeclaredField("mRequestQueue");
        requestQueue = getFieldValueSafely(requestQueueField, network);
    } catch (Exception ex) {
        Log.e(LOG_TAG, "error getting field value");
        return false;
    }
    if (requestQueue == null) {
        Log.e(LOG_TAG, "Request queue is null");
        return false;
    }
    Field proxyHostField = null;
    try {
        Class requestQueueClass = Class.forName("android.net.http.RequestQueue");
        proxyHostField = requestQueueClass
                .getDeclaredField("mProxyHost");
    } catch (Exception ex) {
        Log.e(LOG_TAG, "error getting proxy host field");
        return false;
    }

    boolean temp = proxyHostField.isAccessible();
    try {
        proxyHostField.setAccessible(true);
        proxyHostField.set(requestQueue, proxyServer);
    } catch (Exception ex) {
        Log.e(LOG_TAG, "error setting proxy host");
    } finally {
        proxyHostField.setAccessible(temp);
    }

    Log.d(LOG_TAG, "Setting proxy with <= 3.2 API successful!");
    return true;
}

@SuppressWarnings("all")
private static boolean setProxyICS(WebView webview, String host, int port) {
    try
    {
        Log.d(LOG_TAG, "Setting proxy with 4.0 API.");

        Class jwcjb = Class.forName("android.webkit.JWebCoreJavaBridge");
        Class params[] = new Class[1];
        params[0] = Class.forName("android.net.ProxyProperties");
        Method updateProxyInstance = jwcjb.getDeclaredMethod("updateProxy", params);

        Class wv = Class.forName("android.webkit.WebView");
        Field mWebViewCoreField = wv.getDeclaredField("mWebViewCore");
        Object mWebViewCoreFieldInstance = getFieldValueSafely(mWebViewCoreField, webview);

        Class wvc = Class.forName("android.webkit.WebViewCore");
        Field mBrowserFrameField = wvc.getDeclaredField("mBrowserFrame");
        Object mBrowserFrame = getFieldValueSafely(mBrowserFrameField, mWebViewCoreFieldInstance);

        Class bf = Class.forName("android.webkit.BrowserFrame");
        Field sJavaBridgeField = bf.getDeclaredField("sJavaBridge");
        Object sJavaBridge = getFieldValueSafely(sJavaBridgeField, mBrowserFrame);

        Class ppclass = Class.forName("android.net.ProxyProperties");
        Class pparams[] = new Class[3];
        pparams[0] = String.class;
        pparams[1] = int.class;
        pparams[2] = String.class;
        Constructor ppcont = ppclass.getConstructor(pparams);

        updateProxyInstance.invoke(sJavaBridge, ppcont.newInstance(host, port, null));

        Log.d(LOG_TAG, "Setting proxy with 4.0 API successful!");
        return true;
    }
    catch (Exception ex)
    {
        Log.e(LOG_TAG, "failed to set HTTP proxy: " + ex);
        return false;
    }
}

/**
 * Set Proxy for Android 4.1 - 4.3.
 */
@SuppressWarnings("all")
private static boolean setProxyJB(WebView webview, String host, int port) {
    Log.d(LOG_TAG, "Setting proxy with 4.1 - 4.3 API.");

    try {
        Class wvcClass = Class.forName("android.webkit.WebViewClassic");
        Class wvParams[] = new Class[1];
        wvParams[0] = Class.forName("android.webkit.WebView");
        Method fromWebView = wvcClass.getDeclaredMethod("fromWebView", wvParams);
        Object webViewClassic = fromWebView.invoke(null, webview);

        Class wv = Class.forName("android.webkit.WebViewClassic");
        Field mWebViewCoreField = wv.getDeclaredField("mWebViewCore");
        Object mWebViewCoreFieldInstance = getFieldValueSafely(mWebViewCoreField, webViewClassic);

        Class wvc = Class.forName("android.webkit.WebViewCore");
        Field mBrowserFrameField = wvc.getDeclaredField("mBrowserFrame");
        Object mBrowserFrame = getFieldValueSafely(mBrowserFrameField, mWebViewCoreFieldInstance);

        Class bf = Class.forName("android.webkit.BrowserFrame");
        Field sJavaBridgeField = bf.getDeclaredField("sJavaBridge");
        Object sJavaBridge = getFieldValueSafely(sJavaBridgeField, mBrowserFrame);

        Class ppclass = Class.forName("android.net.ProxyProperties");
        Class pparams[] = new Class[3];
        pparams[0] = String.class;
        pparams[1] = int.class;
        pparams[2] = String.class;
        Constructor ppcont = ppclass.getConstructor(pparams);

        Class jwcjb = Class.forName("android.webkit.JWebCoreJavaBridge");
        Class params[] = new Class[1];
        params[0] = Class.forName("android.net.ProxyProperties");
        Method updateProxyInstance = jwcjb.getDeclaredMethod("updateProxy", params);

        updateProxyInstance.invoke(sJavaBridge, ppcont.newInstance(host, port, null));
    } catch (Exception ex) {
        Log.e(LOG_TAG,"Setting proxy with >= 4.1 API failed with error: " + ex.getMessage());
        return false;
    }

    Log.d(LOG_TAG, "Setting proxy with 4.1 - 4.3 API successful!");
    return true;
}

// from https://stackoverflow.com/questions/19979578/android-webview-set-proxy-programatically-kitkat
@SuppressLint("NewApi")
@SuppressWarnings("all")
private static boolean setProxyKKPlus(WebView webView, String host, int port, String applicationClassName) {
    Log.d(LOG_TAG, "Setting proxy with >= 4.4 API.");

    Context appContext = webView.getContext().getApplicationContext();
    System.setProperty("http.proxyHost", host);
    System.setProperty("http.proxyPort", port + "");
    System.setProperty("https.proxyHost", host);
    System.setProperty("https.proxyPort", port + "");
    try {
        Class applictionCls = Class.forName(applicationClassName);
        Field loadedApkField = applictionCls.getField("mLoadedApk");
        loadedApkField.setAccessible(true);
        Object loadedApk = loadedApkField.get(appContext);
        Class loadedApkCls = Class.forName("android.app.LoadedApk");
        Field receiversField = loadedApkCls.getDeclaredField("mReceivers");
        receiversField.setAccessible(true);
        ArrayMap receivers = (ArrayMap) receiversField.get(loadedApk);
        for (Object receiverMap : receivers.values()) {
            for (Object rec : ((ArrayMap) receiverMap).keySet()) {
                Class clazz = rec.getClass();
                if (clazz.getName().contains("ProxyChangeListener")) {
                    Method onReceiveMethod = clazz.getDeclaredMethod("onReceive", Context.class, Intent.class);
                    Intent intent = new Intent(Proxy.PROXY_CHANGE_ACTION);

                    onReceiveMethod.invoke(rec, appContext, intent);
                }
            }
        }

        Log.d(LOG_TAG, "Setting proxy with >= 4.4 API successful!");
        return true;
    } catch (ClassNotFoundException e) {
        StringWriter sw = new StringWriter();
        e.printStackTrace(new PrintWriter(sw));
        String exceptionAsString = sw.toString();
        Log.v(LOG_TAG, e.getMessage());
        Log.v(LOG_TAG, exceptionAsString);
    } catch (NoSuchFieldException e) {
        StringWriter sw = new StringWriter();
        e.printStackTrace(new PrintWriter(sw));
        String exceptionAsString = sw.toString();
        Log.v(LOG_TAG, e.getMessage());
        Log.v(LOG_TAG, exceptionAsString);
    } catch (IllegalAccessException e) {
        StringWriter sw = new StringWriter();
        e.printStackTrace(new PrintWriter(sw));
        String exceptionAsString = sw.toString();
        Log.v(LOG_TAG, e.getMessage());
        Log.v(LOG_TAG, exceptionAsString);
    } catch (IllegalArgumentException e) {
        StringWriter sw = new StringWriter();
        e.printStackTrace(new PrintWriter(sw));
        String exceptionAsString = sw.toString();
        Log.v(LOG_TAG, e.getMessage());
        Log.v(LOG_TAG, exceptionAsString);
    } catch (NoSuchMethodException e) {
        StringWriter sw = new StringWriter();
        e.printStackTrace(new PrintWriter(sw));
        String exceptionAsString = sw.toString();
        Log.v(LOG_TAG, e.getMessage());
        Log.v(LOG_TAG, exceptionAsString);
    } catch (InvocationTargetException e) {
        StringWriter sw = new StringWriter();
        e.printStackTrace(new PrintWriter(sw));
        String exceptionAsString = sw.toString();
        Log.v(LOG_TAG, e.getMessage());
        Log.v(LOG_TAG, exceptionAsString);
    } 
    return false;
}

private static Object getFieldValueSafely(Field field, Object classInstance) throws IllegalArgumentException, IllegalAccessException {
    boolean oldAccessibleValue = field.isAccessible();
    field.setAccessible(true);
    Object result = field.get(classInstance);
    field.setAccessible(oldAccessibleValue);
    return result;
}

There is no legal way to change your webview proxy settings programmatically. But it's possible to use java reflection to change mProxyHost value from android.net.http.RequestQueue class. It's private value and there is no setters for it, so reflection seems to be the only possible variant. I used it in my project and it works. Here is the sample of my method:

    private boolean setProxyHostField(HttpHost proxyServer) {
    // Getting network      
    Class networkClass = null;
    Object network = null;
    try {
        networkClass = Class.forName("android.webkit.Network");
        Field networkField = networkClass.getDeclaredField("sNetwork");
        network = getFieldValueSafely(networkField, null);
    } catch (Exception ex) {
        Log.e(ProxyManager.class.getName(), "error getting network");
        return false;
    }
    if (network == null) {
        Log.e(ProxyManager.class.getName(), "error getting network : null");
        return false;
    }
    Object requestQueue = null;
    try {
        Field requestQueueField = networkClass
                .getDeclaredField("mRequestQueue");
        requestQueue = getFieldValueSafely(requestQueueField, network);
    } catch (Exception ex) {
        Log.e(ProxyManager.class.getName(), "error getting field value");
        return false;
    }
    if (requestQueue == null) {
        Log.e(ProxyManager.class.getName(), "Request queue is null");
        return false;
    }
    Field proxyHostField = null;
    try {
        Class requestQueueClass = Class.forName("android.net.http.RequestQueue");
        proxyHostField = requestQueueClass
                .getDeclaredField("mProxyHost");
    } catch (Exception ex) {
        Log.e(ProxyManager.class.getName(), "error getting proxy host field");
        return false;
    }       
    synchronized (synchronizer) {
        boolean temp = proxyHostField.isAccessible();
        try {
            proxyHostField.setAccessible(true);
            proxyHostField.set(requestQueue, proxyServer);
        } catch (Exception ex) {
            Log.e(ProxyManager.class.getName(), "error setting proxy host");
        } finally {
            proxyHostField.setAccessible(temp);
        }
    }
    return true;
}

private Object getFieldValueSafely(Field field, Object classInstance) throws IllegalArgumentException, IllegalAccessException {
    boolean oldAccessibleValue = field.isAccessible();
    field.setAccessible(true);
    Object result = field.get(classInstance);
    field.setAccessible(oldAccessibleValue);
    return result;      
}

I've made lots of tests and i can says that the previous response using the override based on android.net.http.RequestQueue works perfectly from android 1.6 to 3.1.

But there was a code refactoring on API and to make it work on Android 3.2 & 4.x, here the solution :

try
{
  Class jwcjb = Class.forName("android.webkit.JWebCoreJavaBridge");
  Class params[] = new Class[1];
  params[0] = Class.forName("android.net.ProxyProperties");
  Method updateProxyInstance = jwcjb.getDeclaredMethod("updateProxy", params);

  Class wv = Class.forName("android.webkit.WebView");
  Field mWebViewCoreField = wv.getDeclaredField("mWebViewCore");
  Object mWebViewCoreFieldIntance = getFieldValueSafely(mWebViewCoreField, oauthPage);

  Class wvc = Class.forName("android.webkit.WebViewCore");
  Field mBrowserFrameField = wvc.getDeclaredField("mBrowserFrame");
  Object mBrowserFrame = getFieldValueSafely(mBrowserFrameField, mWebViewCoreFieldIntance);

  Class bf = Class.forName("android.webkit.BrowserFrame");
  Field sJavaBridgeField = bf.getDeclaredField("sJavaBridge");
  Object sJavaBridge = getFieldValueSafely(sJavaBridgeField, mBrowserFrame);

  Class ppclass = Class.forName("android.net.ProxyProperties");
  Class pparams[] = new Class[3];
  pparams[0] = String.class;
  pparams[1] = int.class;
  pparams[2] = String.class;
  Constructor ppcont = ppclass.getConstructor(pparams);

  updateProxyInstance.invoke(sJavaBridge, ppcont.newInstance("my.proxy.com", 1234, null)); 
}
catch (Exception ex)
{    
}

enjoy


As @Karthik said, these answers are not work on android 4.4(KitKat). For KitKat the answer was posted here.