Can an airport without an FAA controlled tower have their own tower and controllers?
Yes, in fact there are a number of Non-Federal Control Towers (NFCT) in the United States.  These are NOT the same as Federal Contract Towers (FCT), which are funded by the FAA and operated by contract personnel.

NFCTs, on the other hand, are usually funded by the local airport authority.  Some examples (not authoritative or all inclusive): Mojave (MHV), Laughlin/Bullhead City (IFP), and Merritt-Titusville (COI).

Unfortunately, there isn't a consolidated list of all NFCTs.  Rather, they are tracked by their respective FAA region.  Here are the Southern and Western-Pacific Hub Organization Orders that list their respective NFCTs (not sure if they're current).

Operation of NFCTs is covered by AC 90-93b.