It was a cross domain issue. This article here, http://e-mats.org/2010/01/jquery-getjson-and-the-same-origin-policy, explains in detail that you simply add:
?callback=?
…at the end of your cross domain JSON file, and jquery handles the rest. So my URL now looks like:
http://vimeo.com/api/v2/album/1822727/videos.json?callback=?
…and it works perfectly!