Start with these documents in the "Members" section:
Performance Information
Those releases and other membership forms will indicate if the organization is private or public, and if public either County, State or Federal. See who is protected by waivers or releases - is it York County, the Commonwealth, the NPS or The Fifes and Drums of Yorktown (Inc/Ltd/whatever).
As for your comment "It isn't really a "private" organization" - it's either private or public as there's nothing in-between. Organizations referred to as "quasi-public" or "semi-public" are neither. Perhaps the best way to determine status is to see how the folks in charge are selected/appointed - if the government does not appoint them or they do not stand for public election on the same day as other local/state elections the organization is not public. The only other thing you might be able to do to show they are public as opposed to private is to document that the bulk of their funding and the control of their operation are controlled by the government (local, state, or federal).
According to the SCC site
https://sccefile.scc.virginia.gov/BusinessEntity/BusinessEntitySearch.aspx they are not incorporated. You can stop in at their office and ask to see their business license, get their tax ID number, and see other paperwork showing their status as a charity, philathropy, museum or whatever it is they call themselves. They do not seem to have any direct connection to the York County government, per the County's website.
As for "What magic buzzwords make it legal (or illegal) to carry inside the building? " - if they are not a local government agency/board/commission/etc. but private the magic is that they can control behavior on their property pretty much the same as you control it on your property. If they are "open to the public" they are limited in that control so far as not being able to discriminate on the entrance/ accomodation of protected classes, but that's pretty much it.
stay safe.