No, you can not purchase a handgun under the age of 21 as a personal, off duty firearm from an FFL dealer. It would have to be purchased via a private sale. Check your local and Florida laws.
From ATF site, http://www.atf.gov/firearms/faq/lice...-requirements:
Q: Does a customer have to be a certain age to buy firearms or ammunition from a licensee?
Yes. Under the GCA, long guns and long gun ammunition may be sold only to persons 18 years of age or older. Sales of handguns and ammunition for handguns are limited to persons 21 years of age and older. Although some State and local ordinances have lower age requirements, dealers are bound by the minimum age requirements established by the GCA. If State law or local ordinances establish a higher minimum age, the dealer must observe the higher age requirement. [18 U.S.C. 922(b)(1), 27 CFR 478.99(b)]
Q: May a dealer sell firearms to law enforcement agencies and individual officers in another State?
Yes. Sales and deliveries of firearms to out-of-State police and sheriff departments are not prohibited by the GCA. A dealer may also sell or ship firearms, other than NFA firearms, to an individual law enforcement officer, regardless of age, if the dealer has a signed statement from the officer’s agency, stating that the items are to be used in the buyer’s official duties and that the officer has not been convicted of a misdemeanor crime of domestic violence. No ATF Form 4473 or NICS check is required; however, the bound book must be properly posted, and the signed statement included in the dealer’s records. You should contact your State’s Attorney General’s Office to ensure there is no State prohibition on such sales. [18 U.S.C. 925(a) (1), 27 CFR 478.134 and 478.141]