One place I have found very nice in that area is the wildlife refuge north of Yuma. There are 2 or 3 that are along the river north to Blythe,CA. On the West side of the river, it is VERY popular with RVers and you will find neighbors wall to wall. There are a few lakes that are formed off of the river which increase the available camping areas. If you get a vehicle with high clearance, you may find some good camping spots around to the far side of these small lakes.

My favorite spots are on the east side of the river and for the life of me I cannot remember how to get there. I remember a boat launch, and a few camping areas that were lightly used. If there are fisher-people in your group, you will especially like it here.

I hope this helps you with your Arizona camping trip planning. Have fun!

many of the national forest camp grounds in the higher elevations are closed this time of year, so like chuckwagonjack said, you may want to stick to the southern part of the state. you could go to organ pipe national monument, but avoid the established camp ground (it is cramped, and run down looking). there is a primative camp (4 sites) that is first come first serve that is much nicer and very quiet. here is a link for it:

http://www.nps.gov/orpi/planyourvisit/camping.htm

also, in the southeast corner of arizona, you might like to visit chiricahua national monument. i believe their campground is open year around. the chiricahua mountains are quite, not a lot of people go there (one of my favorite areas of the state). here is a link for the chircahua nat monument:
http://www.nps.gov/chir/index.htm

hope you have fun on your trip. also, just to let you know, they are predicting another el nino for us and that means january and febuary could be our wet months. so be prepared for some rain and maybe some snowy storms when you visit.