How does Chantix act like Wellbutrin?

Hi, Rachel!
Unfortunately the two medications contain different drugs. The active ingredient in Chantix is varenicline tartrate but the active ingredient in Wellbutrin is bupropion. The two are not interchangeable. Depending on how long you'll be out of the Wellbutrin you might experience some discontinuation symptoms.
You might try talking to your pharmacist and ask if they can give you a couple of tablets of the Wellbutrin to hold you over until you can get the script refilled.
Best regards and hope this works out for you.