EOD school is at Eglin, correct? No reason for you not to have a firearm local as long as you're legally able to own one. Keeping it in the armory there is no problem... it's staffed 24/7 although they may have specific times you may check in/out personal weapons. There is a really good rod&gun club on Hurlburt with nice shooting range open to you. I'm certain however that you'll have no problem finding somebody to hold your weapon at their home off base if checking it in/out of the armory gets to be a pain. Of course you're gonna get to shoot some way cooler stuff becoming a tech though. The BX at Eglin sells firearms as well.
Unless you're on PCS orders, you can't buy a handgun in Florida unless you're already a resident there. But you can buy in your home-state as long as you meet that states' requirements, and then bring your gun to Florida. But each state has different requirements starting with a minimum age for handguns... how old are you and what state are you a resident of?
You may be eligible to purchase long-gun however. Florida has no law against out-of-state residents purchasing long-guns... they even amended their "contiguous state" clause a few years ago to clarify specifically that you could. However, the state you're a resident of may prohibit buying long-guns out of state... again, what state are you a resident of?
Florida is pretty gun friendly. As long as you're 21yrs old and haven't been convicted of a domestic violence misdemeanor or a felony of any kind, you more than likely qualify for a CCW as military. On TAD orders, you would qualify for a non-resident permit... which allows you to carry in Florida, all the states that border Florida, and a bunch of others. For military, it's as easy as (BEING 21YRS OLD) and walking into your local Sheriffs office with a copy of your orders and military ID. It's $140 total for the permit if you have the Sheriffs office do your fingerprints electronically... trust me, do it that way. Some offices have specific times they do fingerprints, so call ahead.
http://www.usacarry.com/florida_concealed_carry_permit_information.html
http://www.freshfromflorida.com/Divisions-Offices/Licensing/Concealed-Weapon-License
As somebody mentioned, you need your chain-of-command approval to store weapons in the Armory, but not in somebody's house out in town. And you definitely don't need any permission to get your non-resident CCW... you should get it anyways even if you don't plan to carry (as long as you're eligible)... because you never know.
I don't agree with waiting until you get wherever you're going. If you want to buy/own/carry a gun, why wait? The only thing that may bite you is if you buy a gun in Florida that is illegal in anther state. But even then, you can always take an evil-featured gun and convert it to non-evil... and buy lower capacity magazines if that's the issue. I know you can find somebody to trust to hold your gun... HINT: NOT ANOTHER STUDENT... find a damned instructor, maybe even a **GULP** Officer?
As for the BX, I've never ever ever seen a gun for sale in a BX that I couldn't buy cheaper a dozen other places, especially online where there is no tax and lots of places with free or cheap shipping. Also... LGS's in towns with a gun-selling BX will always dicker for cash even if their listed price is higher... walk in with green actual cash and you can almost always walk out of an LGS having paid less money than you would in the BX.