Like so much else in life the answer depends on the details of teh question..... How close can you get to them?
At under 50 yards I'd use a shotshell loaded with any pellets larger than #4 sized shot. So #2, T, TT, TTT, O, OO, or OOO size would do it, but they are all for close range work.
Since I would have a hard time getting within shotgunning range of a coyote unless he was in the hen yard - I'd use a rifle cartridge. I suppose you could use a slug from your shotgun out to a hundred yards or so, but I'd still use a rifle.
I have shot them with everything from #6 shot while out pheasant hunting to 0000 buck. The last few that I have shot have been with T steel shot out of a 10 gauge. All of them will work as long as you are close enough.
My experience with coyotes and shotguns has been limited and not very productive. Both coyotes that I shot was with the "T" shot dead coyote rounds. Both were at around 45yds and I never recovered them. I did have a friend that piled one up at 52yds with #4 buck. My experience with it has made me very leary to use it. But, if I take out the shotty again for yotes, it's going to be a 12ga with 3" #4 buck...
One of the most important pieces of equipment that you will take hunting is a good quality pair of optics and they take some care to give you the performance that you want. So here are a few tips.
Use some alcohol and cheese cloth to clean the lenses. I will usually pick up an extra cleaning kit from my optician when I have my eyes examined.
Don't touch the lens with your fingers. If there is a stick or a bug on them blow it off.
Use the lens covers. They are there...