The state park campgrounds up on the islands are nice
If you want privacy, go to the state park on middle bass. You could then ferry over to the other islands during the day and there's all sorts of stuff to do
Kelleys Island state park campground is nice and there's enough to do there to have fun, and it's more relaxed than south bass island
South bass state park has camping and there's plenty to do on south bass. The bars do get rowdy on certain weekends but the state park is far away from them. There's a little golf course right next to the park that is fun for kids. South bass also has the most to do for kids/families and places to eat
They’re all a good mix of relaxing camping/stuff to do
Indian lake state park has a nice campground
Yogi bear in mt gilead caters to kids/families and has a ton of cool stuff. Not sure how far from you that is but kids would love it