There are different methods to solve the single sign on problem.
If you ensure the cookies are shared across both sites, you can for example write a single sign on handler for eZ Publish that checks which user is logged in with the session cookie in the external app, and vice versa. I don't know about which possibilities vBulletin has there.
About eZ Webin, there is a customization guide here: http://ez.no/doc/extensions/website_interface