The .223 Rem will do the job from just about any rifle on that size target at that range. The ammo is cheap and readily available and you can play with it if you're a handloader. A lot of these varmint rounds are pretty exotic and hard to find...not to mention expensive.
I use the .223 for medium sized game (coyotes, coons and possums) at ranges out to 300 yds or more from a Ruger Mini-14 topped with a NcStar 3-9X scope. I really like the Wolf varmint rounds with the red plastic tip. They feed reliably and expand nicely without causing too much pelt damage. They are affordable and the quality control seems pretty consistent. That's always important with an autoloader. Good luck!
I agree with the above post. The 223 rem will be excellent for what you need to do and the ammo is everywhere. 22-250 may be another caliber that fits your situation for you to look at that is easy to find ammo for. It is more powerful and flatter shooting than the 223. Out of all these calibers, we are not talking about a whole hell of a lot of difference one way or the other.
I am NOT an expert. But I'm too cheap to pay for anyone else to do the job (local shop wanted $200 to tan my coyote hide). I've used this recipe for rabbit hides, deer hides, a moose skin, and a coyote pelt. I've adapted this recipe from one I found online. Feel free to use it but use this tip at your own risk and comply with all local laws wherever you are. When butchering: Cool the hide as soon as you can get it off the animal. Remove the hide form the...