The one thing we have noticed was while we normally stop around 3:00 PM to near empty RV parks they are usually packed several hours later. So if you plan to push the miles and hours on the road I suggest calling in mid afternoon to ensure a site when you arrive.
Good advice.
I do tend to prefer reservations, but even given that preference, I'd still suggest reserving a spot for Devils Tower. The KOA there has an amazing view, just beneath the tower itself, and tends to be popular. Devils Tower is a bit off the beaten path, so there's not much choice in that area. The national park has a CG but most sites aren't too RV-friendly, unless you have a small rig, and it has no hookups.
The drive from Devils Tower to Mt Rushmore isn't long; an easy trip so you shouldn't need an overnight there. You do have a long day ahead of you from Vegas to Yellowstone, though. I doubt you'd need a reservation between those spots unless you'll be there on Memorial Day weekend. Depending what you are looking for, there are several private CGs in the Salt Lake and Provo areas, plus at least one SP (Utah Lake - sites may be smallish) along I-15, so you do have some choice.