You are staying in Cody and then going to West Yellowstone.... ?? Do NOT go to Cooke thinking you are going to find a site in June, they are knee deep in snow up there now, campgrounds are NOT open... plus a long ways out of the way, unless of course you are going into the NE entrance of Yellowstone.
There is a Wyoming State park just outside of Cody that takes reservations. And there are a few other campgrounds in Cody. IF you are going from Cody to East entrance of Yellowstone, there are quite a few campgrounds along the way. Hookups at only a couple of them, but some do take reservations, they are Forest Service campgrounds. Also, a small RV park, nothing fancy called Green Creek RV park by the Red Barn by Wapiti, Wyoming, about 20 miles out of Cody.
Lots of places to stay, just depending which direction you are going. We ran up to a couple of the campgrounds today on the road to East Entrance of Yellowstone today, campgrounds quiet!
Need more info, let us know, lived in Cody 30+ years!
Donna, Mark and Lexie