I-80 has pioneer sites, mostly clustered in central Nebraska near Kearney. Some nice state parks in that area too, for camping, if you're not there on a weekend (they would be more full then). Same advice goes for Starved Rock - nice park, but busy on weekends, likely to be full.
Wow, I'd sure never try to get from Ohio to the Tetons in 3 days. Those are some LONG days. (We used to live in the Chicago area & took 4-5 days to get out here, one way. :) ) With the kids, you'll probably want to pick camping spots with a pool - that time of year it's likely to be hot in the plains - or places with enough space for the kids to run around & play a bit (depending on their ages, of course). Ft Kearney SRA would fit, as would the Holiday Trav'l Park in North Platte. We haven't been there, but some of the other state park "waysides" like Gibbon or Windmill might work for you as well.
East entrance of Yellowstone is fine - the driver will definitely need to pay attention, as the road is winding - but we've done it with our trailer with no issues.