There is a nice USFS CG located at the junction of 89A and 67. The Jacob Lake CG makes a good spot from which to day trip down to the north rim, 47 miles on 67. You can drive to Zion from Jacob Lake in just a couple of hours. Bryce is a couple of hours northeast of Zion.
Your best bet by far for Zion is to go on line and make a reservation for the Watchman CG which is actually inside the park. For just $18 a day ($9 with our Geezer pass), you get nice sites adjacent (and some right on), the Virgin River and they come with 30 amp electrical service. the restrooms have flush toilets and there is a good two lane dump station. Best of all you are just a short walk from the visitors center and the free trams run throughout the park from there every few minutes.
We are leaving tomorrow morning for Zion, have reservations for the entire Thanksgiving week but will of course make the obligatory stop in Vegas on the way up. :B