FL Corporate laws and FL HOA laws would be applicable to proxies.
Per
FL 720.306 (8):
8(a) To be valid, a proxy must be dated, must state the date, time, and place of the meeting for which it was given, and must be signed by the authorized person who executed the proxy. A proxy is effective only for the specific meeting for which it was originally given, as the meeting may lawfully be adjourned and reconvened from time to time, and automatically expires 90 days after the date of the meeting for which it was originally given. A proxy is revocable at any time at the pleasure of the person who executes it. If the proxy form expressly so provides, any proxy holder may appoint, in writing, a substitute to act in his or her place.
Per
FL 617.0721,
2) A member who is entitled to vote may vote in person or, unless the articles of incorporation or the bylaws otherwise provide, may vote by proxy executed in writing by the member or by his or her duly authorized attorney in fact. An appointment of a proxy is not valid after 11 months following the date of its execution unless otherwise provided in the proxy.
(b)
A corporation may reject a vote, consent, waiver, or
proxy appointment if the secretary or other officer or agent authorized to tabulate votes,
acting in good faith, has a reasonable basis for doubting the validity of the signature on it or the signatory’s authority to sign for the member. Based on the emphasized language above, it appears that it's up to the Board. If the Board has good reason to not believe a faxed copy is from the member, then they may reject it.
I would suggest that you ask your Board if they have a problem accepting a faxed proxy from you.