Here's multiplayer chess that I have installed on my site:
http://www.majorleaguetalk.com/chess/
For now, it is a standalone chess server, because it uses it's own user/registration system. It is written in pure PHP and uses a mysql database. There is a phpBB integration for it, but not for vBulletin.
The only real integration with vB that is practical, would be to share the vBulletin User/Registration/Login system and since that's been done for phpBB, it's certainly feasible.
The chess pages were written as a standalone program with it's own css, and getting it to look like the rest of your vBulletin site will be a chore. That' not a real big concern for me, because I don't mind if the chess pages look totally different than the rest of my site.
Registered chess users can challenge other players who, in turn, can accept or decline the challenge. Games can be played at a snails pace and each player receives move notifications via email. If both players are online at the same time, they can switch to a realtime mode where the board is updated every few seconds. Chess clubs can be created, as well as tournaments. Users can customize board colors, and there's a chatbox below the game board.
I've attached screenshots of a game in progress and the admin console.