I am not sure if all of the FS campsites in Alpine Canyon will be closed? Right now both East Table and Wolf Creek are open. It might be worth checking later in the week to see if they close after Oct 1st or with the potential Gov. shut down? The website didn't give a closing day?
Bridger-Teton National Forest - East Table Campground
Wolf Creek Campsite
There is also a lot of camping up the Grey's River Road. There are a couple established campsites with toilets up the road a ways, and lots of undeveloped sites that you can camp at for free. The road is pretty busy with hunters right now, but I found plenty of camping options up there last weekend.