Thanks for that follow-up! It looks like there might be some CSS weirdness going on with that specific site. When using the widget "out of the box", we don't see the same behavior that is on that site. It looks like in order to get it to allow for input, you have to click along what would be the top border of the form. Unfortunately, our ability to troubleshoot custom code is limited, as each site is totally different.
Another option might be to go with MailChimp's "naked" form to the site and style the CSS accordingly:
How can I add my signup form to my website?: http://eepurl.com/gOGN