There's really very little to understand. It's nice that your family owns lots of land but it's quite irrelevent I'm afraid. If you bring a handgun in to NYS without a NYS pistol permit - you will be breaking the law - period. NYS does NOT honor pistol permits from any other state and does not issue permits to non-residents unless you have a very compelling reason like you own a large business in NYS and spend part of the year here. If you are caught with an unregistered/licensed pistol while in NYS - you could be charged with a felony.
p.s. - it doesn't matter if the family here has a pistol permit. The pistol must be listed on their permit for it to be legal. The only way to do that would be to have a gundealer ship it from Virginia to a gundealer in NY who then does the paperwork to add it on their permit. You MUST show custody of the pistol coming into NYS state. You can't just show up with it.