When you do this
var box = $("#show_mailing_list_signup");
box.fancybox().click()
the element <div id="show_mailing_list_signup">
(which contains the form you open in fancybox) becomes the trigger and the target of fancybox so anything you click
inside that container will trigger fancybox over and over again.
Change the selector to something else to target your form like
var box = $(".fancybox");
box.fancybox({
// API options here
}).click();
NOTE: you still need to have an html element (usually a link) with class="fancybox"
to target your form
like :
<a class="fancybox" href="#show_mailing_list_signup"></a>