Had to chime in, although late!
I was lucky enough to live in Jackson. I worked as a live in nanny on Snake River Ranch across from Teton Village. One really fun thing we did with the little ones was to take the ski tram ride up to the top at Jackson Hole Ski Resort. Its a great chance to be way up high on the Teton Ridge without actually hiking it! It is very steep, though. (I am afraid of heights and I was troubled by this.)
The "scenic" float rides are great.
While Jackson can be nuts in summer, June is less crazy than July and August-- and lots cooler, especially at night.
Look into the Teton Country fair schedule and the weekly Rodeos-- a fun taste of the west and small town life that might not be as authentic as it once was, but is plenty fun.
A favorite place to eat for us was Shades Cafe. We also liked the Bunnery. I think they are both still there.
have fun-- I'm jealous!