Furbearer permit is just that. To “bear” or “possess” the carcass (possibly only the “fur”??) of any species designated as a “furbearer” you need to have a furbearer permit. Whether you shoot it, trap it, find it as roadkill, or it drops dead of old age in front of you, you cannot possess the animal unless you have a furbearer permit.
So you could legally shoot it with just a hunting license as long as you left the body in place as a warning to other coyotes, then?