Hi, the problem is that the popup link is inside the popup content, wrapping the complete form. This causes a click on the form content to trigger a new popup, which you see as a refresh of the existing popup:
<div id="contact_form_pop" class="hentry popup" style="width: 460px; max-width: 100%;">
<a href="#contact_form_pop" id="fancybox-auto" class="fancybox-inline">
...
</a>
</div>
Move the link code outside the popup content itself:
<a href="#contact_form_pop" id="fancybox-auto" class="fancybox-inline"></a>
<div id="contact_form_pop" class="hentry popup" style="width: 460px; max-width: 100%;">
...
</div>