We did not find anyplace in West Yellowstone to rent a beacon. We ended up renting from a place in Bozeman, which was all we found.
Panda Sports/Sinclair Gas - maps.google.com
621 Bridger Dr, Bozeman - (406) 587-6280
It was like $25 a day? The gas station opens around 7am and the rental place opens around 8am but the older guy who ran the place was nice enough to get us one at shortly after 7am on our way to West so we didn't have to wait around. They are the older non-digital style.