We are in the process of relocating from Ohio to Arizona. With the process of selling our house and buying one in AZ we are likely to be homeless for one to two months.

We are thinking about living in our fifth wheel for that time. We need a spot with full hookups and decent WiFi for work. The main issue is we have three dogs.  It seems most  campgrounds have a two dog limit. Some also have size limits for the dogs. Our GSD is 100lbs.

Any suggestions for a campground?

check the trap range at Ben Avery Shooting Facility, they have about 100 spaces that fill up in the winter but are empty now. not sure about dog rules, that could be an issue.
https://www.azgfd.com/shooting/bactc/
https://www.azgfd.com/shooting/basf-camping/

McDowell Mountain regional park has an RV campground, again not sure about dogs or length of stay.

wifi will be an issue at either of those place, but you can get some 4G wireless AP from a cellular provider to fill in that gap

Poole

The Ben Avery camp is supposed to be for those participating in shooting events...not sure if they will let you stay there if you are not involved in an event.

I'd sugggest an RV park, but that will get expensive...usually $50-$75 per day or $350-$500 per week.

Spent 10 seconds googling...
https://rvshare.com/blog/top-10-campgrounds/phoenix-arizona/

https://www.azgfd.com/shooting/bactc/campground/

They have an extended stay form. I'd still contact them. The worse thing that can happen is they say no.