I don't know anything about Eglin Air Force base but I'm currently stationed at Ft. Rucker. Ft. Rucker, Alabama has an Archery season that starts Oct 15th and runs until January 31st. Rifle season starts Nov 20th and last until Jan 31st. The limit is 2 does a day and you can shoot 3 bucks a year (all must be 3pt minimum) Anyone can hunt on Ft. Rucker. SInce you are not stationed in Alabama you would have to pay Non-Resident fees. Oh also hog hunting is year round without a limit and Turkey hunting goes from March 15th to May 1st. Your limit is 5 turkeys. There are more Turkeys than you have ever seen. I litterally see turkeys every where I go.
As far as Eglin goes I would check out their MWR as they would have all hunting and fishing info. Tyndall AFB might be anyother good one. This might be better since its closer to you. Basically as far as this type of hunting goes you'll need a game camera to scout for you and figure out where and when they come out. Also a treestand or a groundblind will be needed. Spot and stalking with a floor covered in dry crackly leaves is impossible. I would take to your local game warden, other service members and guys at your local bow or gun shops on tips and tactics on how to bag a deer. Good luck I'll pass on anything I learn. So what do you do for the Air Force at Eglin.