From what I see and read, the part I dont like is the waste and dumping of numerous carcases in a stream. I dont know SD game laws so I'll try not to comment on that.
I have no lost love for coyotes and if they are killed, I dont care if it's a gun or a snowmobile, as long as it is legal. To me, there is nothing more morally wrong in running down a coyote wth a snow mobile than a coyote running down a sheep or calf. IMO, running down a preditor is more "humane" than said preditor(s) running down their game and ripping it apart while it still breaths and struggles. I have a neighbor who lost 30 sheep in one night to a pack of coyotes, so you will see no tears shed from me when I hear about coyotes dying by whatever means. what I dont like about it is wasting them if their hides are worth something, but not really a big deal to me. The biggest deal is the pollution of the stream and the impression it leaves on some toward hunters in general. Kill the coyotes... kill them legal... use their hides if practicle... and dont leave piles of them in view of the general public.
I have never hunted coyote and I don't live in an area where they are a problem. All of what MontanaRifleman wrote above sounds fine to me, but it does raise some questions to a guy like me.
1) Just so I understand, my understanding is that people hunt coyotes for sport or to save their livestock. The pelts aren't worth much and you don't eat them. What ARE you supposed to do with the carcass after a kill? Put it in a garbage bag? Bury it? (I buried my dog a few years back--a LOT of work just for a coyote.)
2) This is an awful and annoying sight that will turn the public against hunting. But ultimately, the real problem is dumping the bodies in a stream. If they were piled up in an open field that would take care of the environmental problem, but the spectacle is roughly the same to a passersby. Does that mean that we are just trying to hide the truth of the matter--that coyotes are usually killed just for the sport of it.
3) In an area that does not have a livestock problem with coyotes, does it still make sense to kill them just for sport? I really don't care personally, but I feel like I have to eat what I kill (or give the meat away) to justify the kill. That might be a stupid way of looking at it, but that's just me.
Just to be clear, I'm not some anti-hunting guy and I don't care if people just hunt coyotes for sport. I also think whomever did this is likely a complete idiot. But in the end, aren't we always left with a pile of carcasses we have to put somewhere?