Yes you can! I did it last year to Yellowstone around the same time. Well, shortly after 4th of July. We were camping, not staying in hotels, so I don’t know what the hotel situation is like. We weren’t able to camp in the park, because it was full when we booked but we stayed in West Yellowstone and didn’t have any issues.
Yellowstone is great for wildlife, we were able to see some wolves while there. But I would also encourage you to stop at the Tetons. There is great hiking and plenty of pull offs to explore.
If you have any questions on Yellowstone, I would love to answer.