IE9 jQuery AJAX with CORS returns "Access is denied"

This is a known bug with jQuery. The jQuery team has "no plans to support this in core and is better suited as a plugin." (See this comment). IE does not use the XMLHttpRequest, but an alternative object named XDomainRequest.

There is a plugin available to support this in jQuery, which can be found here: https://github.com/jaubourg/ajaxHooks/blob/master/src/xdr.js

EDIT The function $.ajaxTransport registers a transporter factory. A transporter is used internally by $.ajax to perform requests. Therefore, I assume you should be able to call $.ajax as usual. Information on transporters and extending $.ajax can be found here.

Also, a perhaps better version of this plugin can be found here.

Two other notes:

1. The object XDomainRequest was introduced from IE8 and will not work in versions below.
2. From IE10 CORS will be supported using a normal XMLHttpRequest.

Edit 2: http to https problem

Requests must be targeted to the same scheme as the hosting page

This restriction means that if your AJAX page is at http://example.com, then your target URL must also begin with HTTP. Similarly, if your AJAX page is at https://example.com, then your target URL must also begin with HTTPS.

Source: http://blogs.msdn.com/b/ieinternals/archive/2010/05/13/xdomainrequest-restrictions-limitations-and-workarounds.aspx

Building off the accepted answer by @dennisg, I accomplished this successfully using jQuery.XDomainRequest.js by MoonScript.

The following code worked correctly in Chrome, Firefox and IE10, but failed in IE9. I simply included the script and it now automagically works in IE9. (And probably 8, but I haven't tested it.)

var displayTweets = function () {
    $.ajax({
        cache: false,
        type: 'GET',
        crossDomain: true,
        url: Site.config().apiRoot + '/Api/GetTwitterFeed',
        contentType: 'application/json; charset=utf-8',
        dataType: 'json',
        success: function (data) {
            for (var tweet in data) {
                displayTweet(data[tweet]);
            }
        }
    });
};