Florida statute 790.23(1)(a) prohibits a convicted felon from possessing a "firearm":
(1) It is unlawful for any person to own or to have in his or her care, custody, possession, or control any firearm, ammunition, or electric weapon or device, or to carry a concealed weapon, including a tear gas gun or chemical weapon or device, if that person has been:
(a) Convicted of a felony in the courts of this state;
Florida statute 790.001(6) defines "firearm" as:
(6) "Firearm" means any weapon (including a starter gun) which will, is designed to, or may readily be converted to expel a projectile by the action of an explosive
; the frame or receiver of any such weapon; any firearm muffler or firearm silencer; any destructive device; or any machine gun. The term "firearm" does not include an antique firearm unless the antique firearm is used in the commission of a crime.
And a crossbow does not fit in the definition of the other prohibited items either (i.e. ammunition, electric weapon or device, or concealed weapon). You can see the full definition section at the following link:
Thus, a convicted felon is not prohibited from possessing a crossbow under these provisions of Florida law.
Please let me know, if you have follow-up questions.