Swanny: I was born and raised in Missouri, moved away when I was 30 in 1989. I hunted near Cook Station, New Haven and Ste. Genevieve, mostly. This year I'll be hunting outside of Cook Station.
To answer your question, YES, AZ has a lottery for most big game except archery deer.
ALL big game firearms hunts (deer, elk, antelope, buffalo, bear) are drawn on a lottery basis. Archery elk, bear and antelope are draw-only basis. Yep, it kinda sucks-like me you may go a couple or few years without firearms hunting.
The plus side is, you can use you favorite deer shootin' iron to kill AS MANY COYOTES YOU WANT, ALL DAY, EVERY DAY, ALL YEAR and the mountain lion seaon (now, they don't have many of those back in MO) lasts all year, tag is over the counter, and is only $13. That should keep ya out in the bush.
To those AZ folks that don't know, to hunt deer in Missouri all you do is buy a tag and hunt wherever you want or can. No draws, lotteries, etc., except for maybe some special black powder hunts in restricted areas. BIG whitetails. (Ft. Leonard Wood)