While Tom is undoubtedly correct that most FFLs will want a driver's license from Florida to prove residency, that isn't what the law says. The BATFE maintains a FAQ site, and this question is specifically addressed there. The FAQ cites the Federal law cited by Tom. What the law says is that, if you own a vacation residence in Florida, whenever you are residing in that residence you ARE a resident of Florida and can purchase firearms in Florida.
You might need to print out copies of the law and the BATFE FAQ and visit a few FFLs to discuss it, but one would hope there will be a way some one of them will be willing to follow the law if you can provide satisfactory evidence that you own the residence and that you actually live there part-time.