CSS flexbox not working in IE10

Flex layout modes are not (fully) natively supported in IE yet. IE10 implements the "tween" version of the spec which is not fully recent, but still works.

https://developer.mozilla.org/en-US/docs/Web/Guide/CSS/Flexible_boxes

This CSS-Tricks article has some advice on cross-browser use of flexbox (including IE): http://css-tricks.com/using-flexbox/

edit: after a bit more research, IE10 flexbox layout mode implemented current to the March 2012 W3C draft spec: http://www.w3.org/TR/2012/WD-css3-flexbox-20120322/

The most current draft is a year or so more recent: http://dev.w3.org/csswg/css-flexbox/

As Ennui mentioned, IE 10 supports the -ms prefixed version of Flexbox (IE 11 supports it unprefixed). The errors I can see in your code are:

• You should have display: -ms-flexbox instead of display: -ms-flex
• I think you should specify all 3 flex values, like flex: 0 1 auto to avoid ambiguity

So the final updated code is...

.flexbox form {
    display: -webkit-flex;
    display: -moz-flex;
    display: -ms-flexbox;
    display: -o-flex;
    display: flex;

    /* Direction defaults to 'row', so not really necessary to specify */
    -webkit-flex-direction: row;
    -moz-flex-direction: row;
    -ms-flex-direction: row;
    -o-flex-direction: row;
    flex-direction: row;
}

.flexbox form input[type=submit] {
    width: 31px;
}

.flexbox form input[type=text] {
    width: auto;

    /* Flex should have 3 values which is shorthand for 
       <flex-grow> <flex-shrink> <flex-basis> */
    -webkit-flex: 1 1 auto;
    -moz-flex: 1 1 auto;
    -ms-flex: 1 1 auto;
    -o-flex: 1 1 auto;
    flex: 1 1 auto;

    /* I don't think you need 'display: flex' on child elements * /
    display: -webkit-flex;
    display: -moz-flex;
    display: -ms-flex;
    display: -o-flex;
    display: flex;
    /**/
}

IE10 has uses the old syntax. So:

display: -ms-flexbox; /* will work on IE10 */
display: flex; /* is new syntax, will not work on IE10 */

see css-tricks.com/snippets/css/a-guide-to-flexbox:

(tweener) means an odd unofficial syntax from [2012] (e.g. display: flexbox;)