We went last year in mid-September. You will absolutely have something to do for an entire week (I argue a week is not enough). The great thing is Grand Teton National Park is only about a 2 hr drive south of Yellowstone, depending on where you stay.Has anyone here ever visited Yellowstone National Park? We're considering visiting next summer, but I'm not sure there's actually enough to do out there to fill an entire week. My daughter will be 6, so I'm also not sure how much hiking she'll be able to do.
Hiking is only one of the things to do at Yellowstone. My son, who was 3 at the time, wanted to hike more than we actually did. But we were limited with a 1-1/2 year old not being able to traverse some of the terrain. A 6 year old should be fine. Get a decent book about the park and it should tell you which trails are easy and which are more difficult.
If you want to see the entirety of the park, it will take you at least 4-5 days just to drive around/sight see. We missed out on some parts of the park due to road closures/snow.
Our oldest son still asks us when we will go back, so I think your daughter should enjoy it. Plenty of wild life, geysers, hot springs, terrain, etc. to keep her interested. You will most likely spend a good amount of time in the car getting to different points in the park, but if you mix that in with hikes and sightseeing, it'll be a breeze.