It's allowed anywhere here. From Mesa the nearest lake is probably
Saguaro Lake. One of the LDS's here occasionally does advanced dives there but mostly they use it for certiification snorkels - not the best viz typically. And boat/jetski traffic on weekends.
Most divers here dive
Lake Pleasant just north of Phoenix. Under an hour drive from Mesa(max) on the 101 freeway from Mesa to just south of Lake Pleasant.
There are no divesites in the Phoenix area busy enough to have on-site tank rentals. Two LDS's in the Mesa area that rent tanks are
EL Mar and
Saguaro Sports. El Mar has an affiliation with the
Diver Down II - the only diveboat in this area which is on Lake Pleasant.
I believe they provide tanks as part of their divetrip.
Here's a link to
Southwest Scuba, since I've used their lake descriptions. Lots of good info on their site about Arizona diving. If you did want to dive Lake Pleasant, one of the other local LDS's - Scuba Specialties does regular non-sponsored fun dives there. Including Thursday night dives at certain times of the year. They have a
forum for people to meet up for dives there.
btw, I
don't lake dive so I'm not the person to contact for details.