Arizona does NOT require work permits for minors and yes there are strict limits on the number of hours those under 16 can work........
in the US under FLSA a minor 14 or older may work in an:
Amusement park, or
Gasoline service station
due to the high the unemployment rate at this time it is very difficult for minors to get a job.........the jobs that use to be held by teens and retirees are now being taken by adults trying to support their families.........
Businesses such as Publix, Kroger's, Food Lion, Sonic, Chic-Fill-A, McDonald's, Winn-Dixie, Krystal's, Dairy Queen, and Burger King along with smaller local businesses that are in the same genre tend to hire those your age.
Fast-food and grocery stores are willing and able to train new employees but........................
Minors tend to have many issues that affect their ability to work such as tardiness, absenteeism, and numerous schedule requests, poor work-ethic etcetera.
I am NOT implying you will have those issues but the fact that many your age do makes it more difficult for you to get an opportunity to prove your worth.
Because of this you must be able to show through words and actions that you are reliable, dependable, punctual, detail oriented, trainable, coachable, punctual, mature and team oriented. You will also need great references since you don't have work experience. Talk to teachers, counselors, clergy, coaches, club sponsors, and directors of places you have volunteered or people you have done casual labor for and ask them about giving you a reference. Get accurate, current contact information for them and ask them to be ready to get calls for references from potential employers.
Most businesses are individually owned and operated so it is the owner’s policy and past experience with minors that will determine if they will hire you. Apply everywhere you can, the worse that can happen is they don't hire you.
For information concerning hours you can work (there are strict limits on when and how many for 14 & 15 year olds), jobs that you can do (there are many others) go to:
US Department of Labor
Industrial Commission of Arizona