I know some places are off limits to casual exploration, Antelope is Indian property and you have to join a tour group. The Wave area is maybe the the one you are thinking about since permits are given out in the morning by lottery. Paria Canyon also requires a permit for overnight camping, here is the
Vermilion Cliffs BLM website.
There are 4WD roads that go into and near the area, Orderville Canyon in particular. The runs from near Jacob Lake to near Kanab. About 30 miles. Plus many 4WD roads around the North Rim area.
I suggest this
book if you want a lot more info