It will be the most scenic riding of your entire trip.
There are hiker/biker campsites at Jenny Lake - plus a camp store nearby.
Dornan's Grill at Moose has a spectacular upstairs deck with views.
Beer, pizza, pasta - etc. Groceries, beer/wine, bike shop.
There are also hiker/biker campsites at nearly every campground in Yellowstone.
The huge, parking lot campgrounds - Grant, Bridge Bay - have pay showers and stores.
(Also Canyon has these services - but it is pleasant with a nice hiker/biker)
I camp at smaller campgrounds and use the facilities at the larger one.
Strongly rec a long loop in Yellowstone - Grant, Lake, Hayden Valley, Canyon, Norris.
Ride super early and late to avoid traffic.
Best - J
PS - I've lived in Wyoming for 20+ years including Jackson.
Photos - Teton Loop & Hayden Valley