Facebook has acknowledged there is an issue with the API (the method VSocial and tools like it communicate with Facebook) uploading videos to a Page.
When trying to upload a Facebook Video to your Page from VSocial, you'll see an error like this:
We are monitoring this closely and will update this article when we learn more. You can follow the Facebook Bug notice here: https://developers.facebook.com/x/bugs/221714001339683/
[Update 25 November 2013] This issue is still impacting Video uploads to Facebook. YouTube video posting is not affected.
[Update 5 December 2013] Facebook has fixed the issue.