You can sleep right in the Dells, and the site is the first booking, not the van. Watson Lake Park keeps dry city sites out on the granite, showers and all, Thursday through Monday nights from the first weekend of April into early October, and you have to reserve at least seven days out (City of Prescott). Lock the night, then pick the van around it. When the city sites are shut, White Spar Campground is 2.7 miles south of the courthouse on Highway 89 and stays open year-round (Prescott National Forest).
Watson Lake is four miles north off Highway 89, and you see the rock best from a kayak, working the lanes between the boulders. Rent a canoe or kayak at the lake, or leave the boat and scramble on foot from either launch. Without a paddle, walk the Peavine, the old Santa Fe rail bed that skirts the east shore.
Lynx Lake, seven miles southeast
Lynx is the pine-lake overnight, a 55-acre trout lake in the Bradshaw ponderosa and the busiest lake on the Prescott National Forest. The campground sits on the west shore with paved spurs about a quarter mile from the water, no hookups, rigs held between 22 and 35 feet, open April 1 through October 31 and booked on Recreation.gov (Prescott National Forest). To get there, head east on Highway 69 and south on Walker Road past Costco.
If you're in something bigger than that, or you want a night on hookups, that's more of an Arizona RV question (RV rentals across Arizona).
Thumb Butte and Whiskey Row
Closer in, Thumb Butte is the hike locals name first, a thumb-shaped rock plug about four miles west of downtown by way of Gurley Street and Thumb Butte Road. Trail 33 climbs it, gaining around 600 feet in a little over a mile to summit benches with the town grid laid out below (Prescott National Forest); it's day-use only, so plan to sleep back at White Spar or Watson. When you come down, Whiskey Row and the Courthouse Plaza are where the evening goes.
Sedona and the Grand Canyon, day drives north
Sedona's red rocks make a real day drive northeast on Highway 89A through Jerome, but the overnight belongs to that market, so plan the van nights around campervan rentals in Sedona. If you're pushing all the way to the Grand Canyon's South Rim, base out of Flagstaff instead, where campervan rentals in Flagstaff covers the trip.