You've got asshole coyotes? Wow, and I thought earwigs* were bad. Of all the things you could have living in your nether regions, coyotes are probably the worst—at least if they were rabbits, you could get some cheap laughs out of changing your name to Warren.
Still, I can't fault your coyotes for eating chickens—I eat chickens myself. (By a similar line of reasoning, I'm also reluctant to censure coyotes for drinking their housemates' whiskey and trying to pay hookers with food stamps.)
As to your plan of shooting the offending animals: The official line is not so much "don't" as "don't bother." You might as well try to eradicate cockroaches with a fly swatter.
Under normal circumstances, a typical coyote pack has only one breeding pair. That means each pack has a half-dozen or more frustrated, horny coyotes only too ready to get their groove on should any holes arise in the coyote population. To coin a phrase: Crunch all you want; they'll make more.
Coyotes don't hunt household pets for food, but they can be dangerous to cats. According to the Portland Bureau of Environmental Services, they may also attack dogs, "especially small, noisy breeds, such as Chihuahuas." (Though frankly, this sounds more like a case of Bowser's mouth writing a check that his ass can't cash.)
Having chickens in coyote country is no picnic, TL, but you're just going to have to suck it up. Experts say your best bet is to fortify your chicken coop to be impregnable to coyotes—and, obviously, don't sign for any suspicious packages from the ACME Corporation.
*Message for those who would point out that earwigs don't actually live in people's ears: Shut up.
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