In Island Park, the best bet is the Henry's Lake State Park - it's a 20 minute drive to the West Entrance of Yellowstone - electric and water. Other options include Red Rock RV Park (on the other side of Henry's Lake) - 35 minutes to the West Entrance and Valley View RV Park - on main highway 20 minutes from the West Entrance. Both have full hookups. If you want to dry camp, lots of FS campgrounds in the area, but most require reservations and are farther away from the West Entrance.
As westernrvparkowner mentioned, if Yellowstone is the destination, there are closer options, but you may have other reasons to stay in Island Park, as there are lots of things to see and do there also. On the Montana side, there is Lionshead RV park and the KOA - both are 10 minutes out, and there are many options in West Yellowstone, with the often-mentioned Grizzly RV Park getting great comments from all. Lost of FS campgrounds on the Montana side also - around Hebgen Lake and down the Madison and Gallatin River drainages.