TN, Mass, Bama all do (I think there are a few more states, but not many), but you must go to a state approved non ABA, i.e. approved by the accrediting body in that state. Please tell me your not talking about an online school, if so, then no only Cali. By "unaccredited" do you mean not approved by the state? I may be wrong, but all non ABAs except for some in Cali are state approved by each specific state.
Some states allow from non ABA grads from other states to be licensed, but typically 5-7 years of practice experience is required.
If you are dealing with an online school check out the distance education board. There are some posters that have decent research on practicing from a non ABA in one state and getting licensed in another state.