WordPress Javascript Widget jQuery Dependency Issue

Because WordPress loads jQuery in no conflict mode, $ is not accessible as a global variable. You need to explicitly use jQuery.

jQuery(document).ready(function () {
    jQuery("#login").loginWidget({
        partnerRef: 0000,
    });
});

If, like me, you want to use $ anyway, you can pass it to the ready function.

jQuery(document).ready(function ($) {
    // you can use $ normally here now
    $("#login").loginWidget({
        partnerRef: 0000,
    });
});

Edit:

Thought it might be useful to add if you want to use a self-initializing function instead of document ready you can pass that jQuery as well.

(function($) {
    // use $ normally now
})(jQuery);