Just came back this week. What a wonderful place and we had a fantastic time. Grizzly is indeed the closest "resort" park to the west gate. We, however, stayed at West Yellowstone KOA. A lovely park that I highly recommend. Trees, indoor heated pool, coffee shop, restaurant, cable TV, and wifi (weak wifi but available). Really a nice place with level sites. We had a premium site with concrete patio, wooden bench swing, concrete round table and benches (really new and nice) grass, and very easy access. If you get down to Jackson don't miss the Bar J Western Show and Dinner. As it happens, we needed emergency RV service on the dash air, Big Sky RV and Rentals were fantastic. We also rented a 15 passenger van from Big Sky, as we were traveling in a group. Wonderful people. We stayed in the West Yellowstone area 6 night, even with some rain every day we could have easily stayed longer and had lots to do if time permitted. We came up the I-15 to the 20. There is about 5 miles of major road construction on the 20, concrete dividers, and no shoulder steep drop off, all along the 5 mile stretch. Big rigs did it, we did it...but I was not happy!!!
Alternate route: We came home going through Jackson Wy, on the 26 to Idaho Falls. This was a good road, no construction, it will take you from Jackson through Yellowstone to West Yellowstone. The drive will probably take more time, but there is no construction (to speak of...as there is always some construction) and the scenery is spectacular.
Happy Trails.