Any USA airport that took Federal Funds through the AIP program cannot be shut down by local government without the agreement of the FAA. FAA letter of guidance attached. I carry a copy right now in case I run into some zealous LEO who might think otherwise. The airports are part of the national infrastructure and might be needed for relief or similar work. Something we all did during Superstorm Sandy up our way a while back. Flying (mainly) high energy food, some water and warm clothing into airports all over the Jersey Shore which was otherwise "closed".
Airports that didn't take FAA funds local government may have a say (especially if it's a town or county owned airport).
However the governor here in RI can have a say as to what happens if you arrive in an aircraft from out of state and step off the airport property - mandatory 14 day quarantine - just like cars coming in from over the border.
That said - when I went to fly my "round" of the 7 public use airports in RI yesterday (see RI is small) - I skipped my usual call at Block Island where the residents are downright trying their hardest to self-isolate the whole island - like the more publicized request of Tangier Island in Chesapeake Bay. Even a touch and go would just plain upset people unnecessarily. I judge to the detriment of GA - even though - sitting solo in the cockpit is about as socially distant as you can manage.
It's a matter of balancing exercise, not getting cabin fever, keeping the engine exercised, and not pissing others off on the ground who think we must be doing SOMETHING wrong.
For al FAA current exemptions to rules caused by COVID-19 see here:
http://fsims.faa.gov/Wdocs/AFS-1%20Memo ... VID-19.htmAnd remember - just because the FAA have made an exception - your INSURANCE company might not. The most obvious being flying on an expired medical. Check with your insurance company to see if you are still insured.