Reply To: Make for appear when you click another item:

[Draz]

Since you have jquery available you could make a simple click handler

This part will make it show up and set focus to it:

$("li.search.icon.menu-item").click(function() {
    $("form.search-form").show().focus();
});

While as a soon as focus is removed this bit will hide it again

$("form.search-form").on( "blur", function(){
  $("form.search-form").hide();
});