I've ran coyotes down on snowmachines before. I know that lots of people chase them down around here when there's enough snow for sledding. When I went I was with a local rancher who also happened to be my basketball coach and he was getting overrun with coyotes on his place. Within 3 miles of his house we saw 8 coyotes from the sleds, but could only get 2 down before they went onto neighboring ranches. We'd chase them for a bit, get them ran down, then park right on top of them. Then as they're crawling out from under your sled you finish them of with a 22 pistol.
Is it fun? HeII yeah. Them buggers are fast and fun to chase on a sled...really makes you a better rider.
Is it ethical? Definitely not. No sport in it at all.
I had a blast doing it personally. I kept the hides(my coach was shocked that I wanted to keep them) and had them tanned. Nice coyotes too. From a management standpoint, you can really do some damage to the population, but you're gonna get some frowns from people who see it as inhumane/un-sportsman-like (which it is). I still prefer to call them in, but to each their own.