Ahh, good sleuthing. I missed that.
When I look at the scripts that are loading on your page, I see the Facebook Javascript API being loaded twice. Once, from us, is loading the //connect.facebook.net/en_GB/all.js file (en_GB is simply the language of your Joomla site), and the other is //connect.facebook.net/en_US/all.js. I can't see the en_US version being loaded as it's probably from a separate Javascript file being loaded first, but that's likely the issue.
Loading the Facebook API twice, especially if it's being initialized with no API key (which would cause the APP_ID value to be blank as you stated) is likely the problem. I don't know where the other load is coming from, but if you can try disabling any other Facebook extensions to narrow it down, that would help. I see the Wibiya toolbar at the bottom as one of the candidates, but there may be others.
Hopefully this helps get you started, and let us know what you find!