I’m not entirely sure how vBulletins login system works, but I have done with with phpBB.

Essentially, you need to look through the login files and see how they store login information (for instance, in a session, in a cookie, etc), as well as what information they store (may be a user name and hashed password, or a user id, etc). Then, you need to replicate that functionality in your own code by taking in information, comparing it to the vBulletin database (with the proper hashing), and then pull the needed information and store it in a cookie/session, or whatever is compatibility.

I know that phpBB implemented a login class that one could simple include to gain that functionality. You may look into that for vBulletin as well. Sorry I can’t be more specific, but hopefully it’ll give you an idea of what to look for.