Home owner associations (HOA's) have the right to limit access to community amenities--(but only for cause like not paying the dues) but even under that regulation they MUST allow access to an owner to his own property. (I think you may get a legal access by right of way over some other land).
Am I right in assuming that you do not live in the HOA, but you have bought a lot that is within the HOA?
Members of the HOA might wonder what benefit a gate would offer to you?
And so is your question, can you build a gate? Or is it that you hope to remove the fence and make your property appear to be a larger "spread" and also maybe appear to be a part of the HOA.
My guess is that you will get a permit for a gate, but that the HOA will not allow the removal of the fence.