I'm thinking the same thing about the HoA....
But I'm also thinking that by the time HoA gets anyone out to ask about what's happening, I'll have everything put away already. Plus, there are no other houses built in my cul-de-sac yet, so I don't think that anyone would raise too much of a fuss.
So there's no actual laws regulating coyote hunting. I believe you can even shoot them at night and you can do so with any caliber you like. With that said, the state law is that you have to be 50 yards or more away from a roadway to discharge a firearm. In addition to that each county may have additional firearm laws. You really should look into the county that you will be living in. For instance, I am about to close on a house in Cumming. The only additional law they have is that you can't discharge a firearm within city limits which is defined as a one mile radius from the city hall.
BUT, be careful of noise ordinances. In Cumming the noise ordinance is in effect from 9pm to 7am every day. If you discharge a firearm during those hours, they can hit you with that.
As for HOA's, from what I understand there aren't many (if any at all) that allow the discharging of firearms. Although you may have everything put away by the time you get there, I wouldn't put it past them to give you a warning anyway or slap you with a fine. Not only that, if you piss off the HOA they will have a target on your house. They'll pick you apart for everything they can find. I can't stand HOA's and I wouldn't want to piss them off if I lived in one. I just don't have time for the "war" that would ensue.
That's my advice. 1) Check the county laws; 2) Be aware of any noise ordinances; 3) Be careful of the HOA.
Otherwise, I'm all about people shooting coyotes. I can't stand those things! Good luck!