Hide a div on webform submission using ajax

Your best option is probably to add an ID to the submit button, or get the ID thats already assigned to it, and stick a bit of jQuery into the header of that page.

(function ($) { 
$('#SubmitButton').click(function() {
     $('#DivYouWantToHide').hide();
});
})(jQuery);

Just replace the ID names with the ones you want to use.

The most effort-free way to hide something i can think to goes trough some CSS.

Inspect the div you want to hide for its id, then just add a rule in your theme CSS

#id { display: none; }