As I read it, NPS can close down the Park, which would effectively preclude tourists accessing the Farm, but should have no voice in how the trust operates the Farm itself. And if the Farm has livestock NPS would need to allow Farm staff to acccess the site to care for said livestock - or suffer an attempt to get a warrant sworn out against them for violating federal animal cruelty laws (which of course then leads to civil suits for compensation for te distress and emotional trauma suffered by staff who knew about the cruel treatment of the livestock).
NPS may get away with keeping tourists from getting in, but telling a lessee you are going to refuse to honor the lease is bordering on the far side of stupid.
But ten, this "government shutdown" is not about shutting down the government so much as it is trying to distract the uninformed and the single-issue voters from the 800 pound gorilla in the middle of the room.
stay safe.