The only range in Eastern Idaho that has long distance capability and is open to the public is the Unified Sportsmen's Club in Rexburg,
http://uscidaho.org/. They are set up shoot out to 1000 yards. Can't say it is a destination point but it sure can be challenging it you like shooting in high wind conditions. Lots of BLM land (the Federal Agency, not the other) in the high desert area that you can set up to shoot much further if you want. However if you are going late in the summer BLM land may be closed for shooting due to fire danger.
How much time are you planning to take for this trip? You are covering a lot of distance and the parks you are going to are not a one day stop and see type. Jackson, Tetons and Yellowstone are a minimum of 2 days with 3 or 4 being much better if you want to see them in much detail. If you haven't made reservations of places to stay, even for camping, it may be too late to find much now.
You don't have Glacier National Park listed to visit but you will be very close to it and it is worth seeing. Lived in SE Idaho for 23 years and have visited many of the areas you want to see. There is a lot of beautiful country out there and driving from one to the other is not quick if you want to take in the sights along the way. So many things to see you really don't want to be in a hurry. This needs to be at least a 2 week trip and an additional week would make it even better.
One additional stop recommendation is the Air Force Museum at Hill Air Force Based near Ogden Utah. Free admission and they have a SR 71 Blackbird on display.