NO you can not carry on FFA property. You really can't even have it in your vehicle. I work at a FAA repair station. It is n the Homeland Security regs. I would leave it in your vehicle but if they ask to search it they can.
My next door neighbor literally is the Chattahoochee National Forest and Lake Blue Ridge. The only thing I can't do on that land is cut down a tree or dig up a plant. I can camp, fish, hunt, shoot without any formal permission. They even have a free to the public boat ramp around the corner from me. They do not allow 4 wheelers or dirt bikes. Camp sites have to be cleared out after 7 days I think but the public camp area with water and electricity has been shut down...so camping up here is actual camping. And of course they strongly encourage cleaning up your mess. It is also illegal here to remove any Native American artifacts found but everybody does it. Hunting is only restricted to whatever season it is and having the proper permit. Otherwise, just pull up park and hunt. FAA restrictions on guns I would imagine has to do more with airplane safety than who owns the land.