When playing with the base game Zombies, is there an official rule for what to do when you run out of Zombies? I checked BGG and the official rulebook and found nothing.
You're right, the official rulebook doesn't cover this, but their FAQ does:
What if I run out of zombies or tokens?
Use any kind of markers you can get a hold of. You can keep track of the zombies you have killed in other ways, such as glass beads, and return the zombies to the pool.
If you want more zombies, Twilight Creations sells a Bag O' Zombies for $10, which I believe contains 100 zombies.