gray tree frogs are so cool! I used to catch them alot on the farm. You can tell there strong bark from quite a distance...that's how I used to find some of them. I kept them for a bit sometimes but not very long usually. Generally I think they would eat common bugs that dwell in trees or like to hang around the yard light at night. If your keeping them through the winter you might try crickets and wax worms.

now to be a stickler: I do believe it is illegal to keep any native species without proper permits. But I don't suspect fish and game to randomly check your pets.