I would think any rifle more powerful than the .22 mag should do the job. A .223 with hollow points should be about ideal but other things suggested like the 30-30 etc. will also kill them like lightning.
I have seen my father kill large hogs with a .22 short but that was at close range and he was a real expert not only with the gun but knowing exactly where to shoot them.
I have a Marlin .22 mag bolt action which I got used for only $90 and it is more accurate than it has any business being, and I think that is common for these guns. It actually shoots 1/4 inch groups at 40 yards.
I put an inexpensive Simmons 4X32 scope on it It should be adequate for coyotes but I would rather have more power.
Any surplus military rifle will be excellent but they all shoot high at close range with issue sights. The Swedish Mauser is sort of the Rolls Royce of surplus because they are both well made and usually extremely accurate and the little 6.5 cartridge is effective against any thing inluding elephants with the right bullet. To top it off, the recoil is not too bad.
Another handy gun which would also be effective would be one of the modern Wincheste 92 copies chambered in .357 mag or .44 mag. (or anything it is regularly chambered in, for that matter). It would also have the advantage of not having an extreme range in case you miss high.
When Woolworth’s was still in business I purchased two of those Swedish Muasers for $60 each. They had barrels of them along with Garands.
Mossburg 500 or Remmy 870 (Look at the pawn shops and used guns at dealers).
I say go with 12 gauge and use Remington Controlled recoil buck shot ( or Hornady TAP ammo).
You then have a gun that can do just about anything from rabbits to T-Rex with the proper load.
“12 gauge - accept no substitute”