It really depends on which permits you mean. If you need a permit for a particular species, you really really had better get the permit before you get the animal, because if you get caught, there's the fines, the seizure of not only that one animal but possibly more, restrictions on what you can own, and no chance whatsoever of getting the permit then.

So it'll depend on what state, what animals, what permits, what permit system, and how lenient the local officials are about you owning the animal BEFORE you get a permit.