Do people have allergies in Phoenix? Yes, they do.
Some people come to the desert for relief from allergies. You'll find people who will tell you that their allergies got worse, and some who'll tell you that their allergies got better. Some people never had allergies before, but suffer with allergies after they move to the desert. Some don't. I don't believe that there is any empirical data that will help you to determine whether you will or will not have allergies in the Phoenix area. If you do end up with allergies, you won't know how long during the year you'll be sniffling or how severe your allergies will be. If other considerations are not overriding, locating in a lesser populated, rural part of Arizona may afford you a better chance at relief than in a high growth, densely populated area.
So what is causing so many people to have allergies in the desert? I have three nasty words for you: pollen, dust, and pollution.