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...
For most on here there is a dream trip floating up around in their heads but there is a question of how to pay for it. This has been a problem for all hunters that wish for that elusive moose, bear, goat, sheep, elk, deer, or any other hunt where you are required to hire a guide or you choose to hire one for that hunt of a lifetime.
First off I am not a financial planner I am just a hunter that learned a long time ago that even with a great job there are going to be some...