I need to add a new database to my website that can sort names, stats, and so on for members, they would log in and edit/enter this info themselves. However the main part of my site is the vBulletin. So is there a way to use their existing vbulletin login/password to also grant them access to the new database/page I'm going to create?
yes.
You're going to have to be more specific if you want to know how it can be done. But it can definately be done. Everything can be done... It's always a question of how.