Need to buy a rifle to hunt a coyote.
You mean a canine coyote , right?
Know local law regarding license required ,if any. As well as any hunter safety requirements. And safety zones near occupied buildings if they exist .
Your set up /location will determine what you need.
A shotgun will work on some sites.
Where pelts are prime (a limited window) and you would put them to use a smaller caliber usually gets the nod. .204 , .223 , 22-250 , and there are others.
If not wanting hides being a priority , you have many more options.
What rifle(s) do you own now?
Will you be calling coyotes?
Will you be shooting them at night?
No sense buying a rifle for one coyote. Invite experienced coyote hunters to do the work. If you blow it , a coyote just becomes more educated.
For more effective coyote control , get a trapper.
Coyotes though respond to successful persecution by creating more coyotes.
The reason you feel a need for a coyote rifle may have other angles to work for when you are not around but a coyote or coyotes is/are around.
I'm all for rifles , don't get me wrong. But a dedicated coyote rifle vs a more utilitarian rifle can chance spendthrift.
If you have a long range proposition and do not hunt other critters with experience , it will require range/target practice time... and study of coyote behavior.
Seeing one regular does not mean it will be a cinch to shoot it. And it may indicate multiple coyotes in your locale.