June should pose no problems with a car. You'll be on the interstate except for a small portion of your trip.
When you take I-15 to Idaho Falls then east on 26, if it happens to be snow (I doubt it), don't take 31 to Victor and on to Jackson. It's Teton Pass and very steep and wouldn't be good in snow. If it's dry, definitely take it as it's a beautiful drive otherwise.
Instead, take 26 to Alpine, WY then north on 26/89 to Jackson. It's a flat drive.
That time of year will be gorgeous as the mountains will still have snow - toward the end of summer it's mostly gone. Hope you get to go. It will be worth it!