Most states have an open season when it comes specifically to coyotes (shoot on sight), but there may be restrictions on night time hunting and the use of lights. A hunting license isn't even always required when shooting on your own property, but the locality may have laws about firearm discharge. Coyotes are overrunning much of the areas in which they are found with extensive damage to wildlife and livestock because not nearly enough are harvested to keep their numbers in check. Before you go after any predator, ALWAYS CHECK YOUR LOCAL LAWS FIRST.Maybe some type of population control permit or the law could just allow it. Some states do.
Predator hunting has really picked up popularity in my area in the last 5 years. There are clubs that put on competitive hunts.
I am a hunter and therefore a conservationist. No hunter will tell you "we need more coyotes." If you have any hunting friends, enlist their services. I'd do it just for excuse to continue hunting.
Indiscriminate killing (poison) is NEVER a good option. Trapping and relocating is an ongoing operation that will never end unless the predators are exterminated (also a terrible idea). The best solution is that which you have the most control, prevention.
I free range chickens in a VERY predator rich environment and lose only 1 bird a year (acceptable loss). Deter daylight attacks and keep a secure coop at night and you'll be fine.