What SixGunner said. The non-resident Class G (general) hunting license will cost $113.50. Resident licenses are $25.50. I'm not sure what you need to prove residency for a license, but anyone resident more than 30 days meets the residency test for buying a handgun. Utility bills with a physical address might satisfy the requirement, as might a housing lease of some sort.
A Class H non-resident license, good for 3 days, will let you hunt coyotes. They cost $51.50.
Licenses can be bought over the counter at WalMart. There is a WalMart at the east end of Sierra Vista.
Most of the ranchers in southern AZ will be happy to let you hunt their land for coyotes. Most ranches are not posted at all... with AZ hunting laws, any private land not specifically posted "No Hunting" allows hunting. Land posted "No Hunting" requires the written permission of the land owner before hunting the property. Some ranchers put signs on their property requiring hunters to sign in on the book attached to the sign before hunting as a way of granting permission to hunt there.
Two good spots near to Sierra Vista are the Lindsey Ranch at mile marker 12 on Charleston Road (the road connecting Sierra Vista to Tombstone). The gate stays closed but if it is not locked you are allowed to hunt there as long as you honor the restrictions posted on the sign at the gate. Close the gate behind yourself on the way in or out.
The other is the Cowan Ranch just east of mile marker 1 on Gleeson Road (just east of Tombstone). Sign in at the box on the signpost where Cowan Ranch Road leads south off Gleeson Road.
There are plenty of other good spots around SV, just ask around.
http://www.azgfd.gov/h_f/game_coyote.shtml
