No one on an online forum is gonna be able to tell you a straight "yes or no" because we don't know your situation.
Here in in redneckville, it's legal to shoot nusience animals (raccoons, possums, whatnot) and that's not considered "hunting" says so in the game and fish hunting manual what is and isn't legal
SO you should probably at least be aware of any wildlife regulations for your area before you go out guns a blazin.
of course there are laws about shooting in the city too, but you didn't ask about that
check your state hunting regs. Most places right now it is season for furbearers, so if you get a liscense you should be covered (as long as you aren't in town and breaking the shooting in town laws). If something is killing you livestock, you are generally OK, but again check your state and local laws.
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Yes which in some states may or may not be exactly the same as hunting.
You need to know your state wildlife laws, federal wildlife law, and your local ordnances. Not knowing is no excuse and in all likely hood will land you in the big house with the barred windows and razor wire fencing.