I'm just quoting the person on the phone. To their credit, by "real lab" I did not get the impression that they meant it had to be a fully equipped lab - they were talking to me, a homeschooler, and they were trying to be helpful; I did not feel they were being obstructionist. I got the impression they just meant it wouldn't do to do a few experiments with measuring cups and spoons and call it the equivalent to being able to identify a beaker or Erlenmyer flask and that we should use some chemicals besides those easily available in the kitchen. Is that fair? Well, if you have a really well designed kitchen chemistry program, maybe not. And maybe they will accept it if you explain it well enough. My impression was just that they wanted students to be comfortable about what lab equipment consisted of, how to measure, how to be safe and how to record before they allowed them in the lab there.
As for whether every school in AZ has a fully equipped lab -- I don't know. I doubt it. The high schools in Tucson and Phoenix and their suburbs that I've seen have good ones. Whether every little district out in the desert or school on all Native Lands do, I very much doubt it.