Arriving in the afternoon, the chance of a Vacancy at Atlatl CG is probably zero. We just returned from S Utah, where it is incredibly busy. We visited Capitol Reef NP, but didn't even attempt Arches, Canyonlands. Bryce or Zion. There are so many more tourists in the wake of Covid. Do you have park entrance reservations for Zion? I heard reservations are required for some NPs now, Zion being one of them.

Not really sure about alternatives to camping if you don’t get a spot. Maybe Boondocking opportunities to the west of the park before you enter or campsites skirting Lake Mead. To walk to the visitors center is closer to 3 miles, not one as someone else had indicated. You would definitely want to do that early morning as by mid-June mid-day temps will already be in the 90’s, if not the hundreds. Atlatl rock is close to the campground and has petroglyphs on it. There may some rock scrambling opportunities close to the campsites. But the “real” trails are by the visitor’s center and the roads beyond it. Pretty far to walk just to get to, nevermind the trail itself. Will you have bicycles maybe? If you pedal, have lots of water!

The temp will be over 100 likely so no on the hiking. There are other rv areas, Poverty Flats etc and BLM land, but no electricity.

You are in the park. Walk from your RV around the rocks and walk over to see Hyroglyphs. If you are there for the day I think there's enough to do right there.

Atlatl is within the park, but the park is quite extensive and you’d need a vehicle to get around. No shuttle, nothing else nearby. There are campgrounds at Lake Mead, which is nearby, and boondocking at a BLM location west of the park entrance, but again you’ll need to drive to and from them.

I am not sure about surrounding parks but you can boondock outside the state park entrances. There are plenty of areas to do that from the east entrance. In my experience, you will not get a space after 9:00-10:00 am. People pack up and head out early and there are people scoping out spaces early. From the campground, it is an easy walk to the petroglyphs though. Good luck.

You should have no problem getting a spot at Alta Rock at that time of the year. Temps will be 100 . The campground is about 1 mile from the visitors center and hikeable, but again 100 temps. You will need a vehicle to get to all the sites within the park (all within 8 miles of the campground. No shuttle service available within the park. All the sites to see within the park has good parking for a motorhome. That would be my recommendation. The campground is very nice and worth staying at. Mountain goats go through the campground a couple times a day. And the best petroglyphs in the park are about 300 yards away, a very easy hike. No other rv campgrounds near by, but softer leaving the North entrance of the park there is dispersed BLM camping.