It's not exactly the same as a residential lease. Depending on the state (and of course how the lease is written), there are specific rules on leases for agricultural land. One common setup is that the tenant retains ownership of the crop and the right to tend to it and the lease expires the moment he harvests. So if the club has a hunting lease that says 'for the 2018/2019 season' it may not magically disappear just because title to the land has changed hands. It may go 'poof' with the sale, but I wouldn't bank on that.
Get a copy of whatever paperwork the seller executed with the hunt club before you start removing tree stands. If he doesn't deliver, start sending lawyer letters citing whatever parts of the purchase agreement for the land that apply to the lease issue.
That is exactly the way it worked in Cecil County when my brother bought his place. The existing lease continued until the growing season was over and the farmer harvested his crop.
OP, late firearm season is this Friday Saturday, and in some locales, Sunday. Keep an eye out for hunters and let them know the property has changed hands.