I do not practice in AZ, but, because no one has answered yet, I can give you some general advice. In my experience, people on SSI usually qualify for public defender assistance. The likely reason she was denied a public defender is she has too many other, valuable assets. You stated she has a car. If she owns a car, that alone would be enough to disqualify her in WI, as she could sell the car to pay for legal services. (No joke.)
To find out specifically why your daughter was denied, you should contact the public defender's office that rejected her.
One thing is for certain: Your daughter needs an attorney. If the public defender's office will not provide her with one, she should determine if the court will appoint a low cost attorney to her case. If neither of those options work, she needs to do everything in her power to hire a private attorney. Some private attorneys do take payment plans.