I have vacationed in Yellowstone the past 2 summers with my family and have in the park at least 6 times.
Having bikes in the campground is very handy. Otherwise there is almost no MTB riding in the park and I would recommend against road riding. The traffic is really heavy in June. There are large numbers of people who rent RVs and tour the park - many have no experience driving a vehicle any larger than a sedan - many may drive on the other side of the road in their native lands. Combine this with narrow shoulders and it is very scary. I did a 65 mile day in the park last summer - and I wont try that again - due only to the traffic. I also did a road ride from Teton Park to Jackson and back, 75+or-. Again, with only 1 main highway (2 lane) between Jackson WY and Yellowstone, traffic was an issue.
Don't forget that the elevation in the park is significantly higher than FL and there is a lot less oxygen
Most of the sights in the park are far enough apart that family riding is probably not an option.
Yellowstone is a GREAT
family vacation spot and we will go again, soon. Enjoy it truly is an amazing experience.