possibly take one out with a flashlight and a Remington 11-48 12 Gauge with buckshot.
I've done limited night time hunting for coyote and from my experience you'll be lucky to keep them within shotgun range if you put a flashlight on them.
Get yourself an AR or other semi-auto rifle with a large capacity then sit about 100 to 200 yards away from the kill zone on a night with lots of moonlight. Wait for them to come in or better yet set up a call box then when they come in turn on the flood lights and light'em up with your rifle.
Here in WI you can only use a flashlight to shine coyote and raccoon AT THE TIME OF THE SHOOTING. Stupid I know but its how the law is written. Using that method though my buddies and I sat with his work lights set up on his hill behind his house overlooking a small gully. Using calls we brought 3 in and upon switching the lights on we took all three out at about 250 yards. It took 15 rounds from me and 10 from my buddy but we got all three. The first one down was hit and dropped with the first shot (mine) and my buddy and I opened up on the other two. No idea who got the last two.
My point though is that even at that distance once the lights went on they turned and fled. If you manage to get them in close enough for shotgun use you'd better be able to shoot it without a light. Turn a light on and the yote is likely to be gone.