I live in Pinedale which is about 2 ish hours from Yellowstone and have pretty much fished all around Yellowstone. It all depends on where you and your family are going to be. Are you going to be in West Yellowstone close to Island Park Id, south Yellowstone Jackson Hole Wy and Teton Park area or on the Cody Wy side of the park?
If you are on the west side and have access to a boat, I would recommend Henrys Lake about 15 miles from Island Park. It’s really late in the summer so the fish are going to be pooled up in and around the springs and creeks that flow into the lake. I was just there last week and we did anywhere from 20 to 30 fish a day in Targhee creek inlet using a copper john and a mity mouse trailer on a strike indicator, limit is 2 but we were catching and releasing. My brother-in-law landed a nice 6 lb and 9 lb hybrid trout.
If you’re going to be close to Jackson Hole I would fish the upper Snake River and Hoback River where the water is a little cooler. Those rivers will produce some nice browns, rainbows and brook trout.
As for the eastern side close to Cody Wy, I’m not sure what the name of the river is but it flows into and out of the buffalo reservoir. Being this late in the year it’s a great dry fly river.
Hope this helps!
Here are some local blogs that show some great lip ripping!
www.awbigfish.blogspot.com/
www.eric2rivers.blogspot.com/
www.fishindog.net/