Well prepare for sloooww traffic. You will be lucky to make 20 mph on average in the park. There are many traffic stoppages when people stop in the middle of the road whenever they see a chipmunk. As mentioned above you are rolling the dice for weather that early, YNP is at 8000 ft so winter can hang on.
There are lots of people around all the thermal features, geysers and such. And around the Yellowstone canyon. The east side of the park can be somewhat empty in comparison but does not have the spectacular features. But it does have critters, we saw Grizzles over there.
We've been up there 6 or 8 times, last summer briefly while we spent 4 days at Tetons. We spent 8 days twice and still haven't seen the whole park by a long shot.