Yes, they must give you keys to the gates to walk in/out of the community.
When you purchase property within a community, you also become owner the purchase of your unit comes with an easement to for the ingress and egress into the community.
You should become aware of the HOA's governing docs (Declaration, Bylaws and Articles of Incorporation). You should've been provided a copy when you purchased your home. If you do not have a copy, you can obtain a copy from the County Recorder's Office (in Miami-Dade County they are available online - see link below). Within these documents you should find the language that grants you, as an owner, the easement mentioned above.
Notify the Association/HOA, preferably in writing, that they are required to provide you with gate keys, and cite the appropriate governing docs.
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Fla. Stat. 720.305(2)(a), which deals with fines and suspensions, states you must be given vehicular and pedestrian ingress and egress to your property. If they have to do it if you are breaking the covenants, they surely have to do it if you're obeying the covenants.
(2)(a) An association may suspend, for a reasonable period of time, the right of a member, or a member’s tenant, guest, or invitee, to use common areas and facilities for the failure of the owner of the parcel or its occupant, licensee, or invitee to comply with any provision of the declaration, the association bylaws, or reasonable rules of the association. This paragraph does not apply to that portion of common areas used to provide access or utility services to the parcel. A suspension may not impair the right of an owner or tenant of a parcel to have vehicular and pedestrian ingress to and egress from the parcel, including, but not limited to, the right to park.
This communication is not intended to create an attorney/client relationship. It is always recommended you consult an attorney in person to discuss your case.