It is possible to kill coyote with a shotgun. And buckshot. A slug would kill it, if you can hit the thing with one (although you might recover the nose and tail after hitting it with a 1oz slug).
For buckshot, you'll want a full choke. And preferably a long barrel. Couple years ago I got to use an Ithaca 37 Featherweight in 12 gauge. I was a little unsure of how well it would work, being a full choke and a 28" barrel. But when a guy I was hunting with chased a tree rat up a tree (and missed twice, causing the lil guy to come around my side of the tree), I decided to give 'er a try. I wouldn't have tried with the 870 I had been using with a 24" barrel and choke tubes, but I suspected the Ithaca might reach out the 75+ yards..... and it did. My hunting partner was not amused, but I was impressed.
But often you will not see coyote close enough to whang away at them with a scattergun and score hits.