How to sequentially run promises with Q in Javascript?

but instead they are being output immediately for some reason.

You are calling them already here:

promiseFuncs.push(getDelayedString(str));
//                                ^^^^^

You would need to push function(){ return getDelayedString(str); }. Btw, instead of using pushing to an array in an each loop you rather should use map. And actually you don't really need that anyway but can reduce over the strings array directly:

function onceUponATime() {
    var strings = ["Once", "upon", "a", "time"];

    return strings.reduce(function (soFar, s) {
        return soFar.then(function() {
            return getDelayedString(s);
        });
    }, Q());    
}

Oh, and don't use document.write.