Yellowstone is overcrowded, overrun with foreigners and tour busses this time of year. Also it takes 2 or 3 days to see the park, it is a long ways between the attractions and there is usually construction in the summer. Also anytime anyone spots any wildlife the traffic really backs up. I enjoy the drive on the mountain roads in the area more than the park itself. The small kids will quickly be bored.
This. I hate to be a downer, but he's right. I was there 4 weeks ago, which was still pretty early in the season, and even then it was quite busy. Definitely big tour buses, definitely stupid, stupid people stopping everywhere, limited parking. And yes - construction. It is a beautiful park, and maybe you have unlimited patience, but I don't, and with 2 little ones, I'd simply not go there this time of year.
The road construction is on the NW section of the Figure 8 "Loop Drive" still in the park (As if you were going to exit on Rt. #89, heading N. to Gardiner). I'd
seriously suggest avoiding this section, as it's major road construction and 4 weeks ago traffic was seriously impeded. The normally 2 lane, is down to lots of one-way rutted shitty roads with dump trucks, construction equipment, buses, etc. Ugh!
Unless you've reservations, you'll not find lodging or camping either. Believe me, it's a zoo. We had 2 full days, with moderate traffic, and it was not really adequate to do justice. You're going to spend a lot of windshield time in traffic. Sry man, but be prepared. This park is way over capacity this time of year ... way, way over capacity.