Only one game in town: Chabad of Jackson Hole. Rabbi Mendelsohn is extremely warm and visitor friendly and will definitely have a Shabbos minyan in August, possibly even weekday if there are enough tourists. But lodging in town is VERY expensive, so book ASAP.
Also know that Yellowstone is currently closed due to flooding, and even though parts of it will almost definitely be reopened by then, the northern part probably won't be, which will mean even more crazy traffic and possibly even timed entry to keep crowds under control.