My opinion,... either of the 2 will suffice.

But if it were ME at that distance,.. a good ol' 12 ga. in 00 would do nicely! There'd be NO escape !!!

The heavier mass of the 22wmr is the better cartridge. 17hmr will work if you're holding it to close in shots. There are three 17hmr loads I would consider for coyotes:

CCI 20gr gamepoint softpoint
CCI 20gr FMJ
Hornady 20gr HP

The advantage of the 17 HMR is accuracy out past 100 yds. The disadvantage is the light bullet though fine for rabbit or squirrel size game doesn't quite have the heft to effectively or consistently anchor yote sized game. Oh sure it can do it if everything aligns right. But, in your case and out to 100 yds I would rather be packing the 22 Mag with a heavier 40 grain or I believe they even know make a fifty grain bullet for it. Think 50 grains smacking that dog as compared to 17 or 20 grains with the 17.

Why do you guys shoot coyotes? do you eat them? hide? claws? Just curious.

In the north, they have nice hides that you used to be able to sell.
In the south, it's cause they kill calves and sheep.
Also, people get bored in the off season and coyotes are open year-round.

I don't think anybody tries to eat them.

Actually, some do eat them. Most don't and like you said hunt them for their pelts, but we've had discussions about eating them and some do. There are recipes out there for coyote.

I see, thanks.

I've only seen one 17 and shot it twice at a target. Have no idea how it would work, awful small bullet. I've used the 22 mag to kill a number od large dogs chasing stock. It works very well.