Does the post office deliver on Saturday in Arizona?

Yes, it does. Almost all non-business, non-rural, non-contract deliveries are made on Saturday. Career city carriers work 5 of 6 delivery days, with about 1/6 having a Saturday off and the other 5/6 scheduled to work on Saturdays. As of 2016, the delivery of some Priority Mail has been extended to Sunday delivery as well, and Express Mail delivery is available 365 days a year.