You picked a nice time of year to visit here.
The closest dive shop is Shark Quest, it's about 15 mins west of Navarre on HWY 98. I have bought gear there and get tanks filled there but I have never been out on their boat. I think they said they do lionfish trips but I'm not sure where they leave from. There are no boats that leave directly from Navarre.
The closest boats are on Pensacola Beach to the West and Destin to the East.
If you go East, Scuba Quest in Destin is pretty good, and West I'd go with MBT in Pensacola. I think both shops can setup lionfish charters.
Good luck.