Definitely make reservations for a campground inside Yellowstone. You absolutely *CAN* tent camp--and btw, the water is fabulous. It comes from a deep well and its delicious & safe to drink.
We camped at Bridge Bay last summer in a trailer. Its a large campground--try to get in the back, if possible. A lot of the trees were cut down in the front, so no shade. But the back is a little more inviting. They have bear boxes to lock up your food-- and know this: the rangers take food seriously! You cannot leave any food or food-containing equipment in your campsite, not in your tent or by your car. And no sunscreen, toothpaste, lip gloss, deodorant, or anything else that has "scent" in it. If they drive by your site and see a trashbag or cooler or water bottles or a camp stove which is not in use, they WILL stop and talk to you. If you are not there, they will confiscate the stuff they find. Don't test the ranger!
I don't htink you'll have any worries about staying in the campgrounds. The animals don't usually want to hang out with visitors. You'll see them at times--be sure your kids understand that they are NOT tame. Buffalo and elk can hurt you real bad! Stay far, far away from animals. You probably won't see a bear unless you're lucky. Don't let the kids go off unsupervised though.
Have a great time. We love YNP.