I recently built an AR-15 specifically for a coyote gun. I had mine chambered in .223 Wylde with a 1:8 twist, 18 inch barrel. I chose that caliber because I have a ton of .223 brass and I am reloading for it. You can load .223 Wylde a little hotter than .223 Rem, but still use standard AR-15 parts. I put a light, crisp, CMC trigger on it, which I love. That's probably the 2 most important pieces of the build, the barrel and the trigger. For optics I put a 4-12x50 Vortex on it. The reasoning for that scope is for a wide field of view, with just enough magnification to reach out to ~500 yards. I wanted to be able to quickly find a running yote in the scope. I also opted to get a Tegra Arms carbon fiber lower receiver for this build, as it doesn't get near as cold to the touch in the winter.
If you have any inclination to suppress it, I would recommend a .300 Blackout shooting subsonic ammo. Ballistics aren't near as good, but you gain so much by having it suppressed when hunting coyotes.
As far as makes and models, there are a ton of good options. Primary arms is a good source with good prices. If you are new to building them, Brownells has a paper you can download that shows you exactly what you need to buy, whether it be by individual piece, or by subassembly (ie BCG vs bolt + bolt carrier, etc.). Palmetto State armory is another good website for AR parts.
I realize I'm a little late to the conversation, but that's my 2 cents anyway. Good luck!