First things first you need to check your state laws. Second Coyotes are not the eaiest animals to hunt, not all of them are as stupid as the cartoon yote. An AR will work just fine in 223 it will do a good job. Like some other said i know guys that just grab a gun it could be a 223 or a 7mm RUM or a 12 ga with 3" 00 Buck loads. Yotes don't really care whatyou blow their guts out with.
I would reccomend you go over and visit http://www.predatorhunters.com/
These two sites deal mainly with whackin yotes. Be carfule on predator masters some of them get a little pissy with new guys and some get pissy if you hunt yotes at anytime other than Nov-Feb.
You are also going to need some calls. Mouth calls are the cheapest but take time to learn. electronic callers IE FoxPro are expensive ranging from $100 on up to $700+ for the top of the line fox pro full of sounds.
I am going to sujest that you get yourself some Randy Anderson
videos. Randy is the undisputed king of yote wackin. Les Johnson and his brother
are another set of yote smackin gods as well. They have a tv show and they put out some DVD's as well. You can learn a lot from both.