Fishing Bridge is the only campground with full hookups within Yellowstone NP. The sites are very close together. You will get to know your neighbor. Some of those sites can be a chore to back into. All the other campgrounds in Yellowstone donít have any hookups at all.
Just outside the West gate of Yellowstone there are two RV parks in the town of West Yellowstone, MT that offer full hookups. Both of these parks, Yellowstone Grizzly RV park and Buffalo Crossing RV park are a bit pricey. Both parks are within a block of each other and both are just a couple of blocks from the west entrance. About 7 miles from the west gate there is also the West Yellowstone KOA that gets good reviews. There are other campgrounds in the town of West Yellowstone but I didnít care for them. You can read reviews of these parks by clicking on this link. West Yellowstone, MT Campground Reviews - Best of West Yellowstone Camping - RV Park Reviews
Other full hookup choices are in Gardiner, MT just outside the North gate of Yellowstone NP. The two RV parks are Rocky Mountain RV Park and Yellowstone RV Park. You can read reviews of those two parks at this link Gardiner, MT Campground Reviews - Best of Gardiner Camping - RV Park Reviews
A few miles from the South gate of Yellowstone NP is Headwaters Lodge and Cabins at Flagg Ranch. The park is under new management since we stayed there so I canít recommend this park one way or the other. You can read reviews at this link. Moran, WY Campground Reviews - Best of Moran Camping - RV Park Reviews
As dj3spots mentioned, there is a KOA in Cody, WY along with others but Iím not a fan of wasting 2 hours of driving time every day to and from the park.