Use whatever weapon you have. Here in Texas, we are allowed to carry a sidearm, and I used a .22 pistol to dispatch my son's first doe, after he shot her a little too far back and a little too low.

I guess whether to finish a wounded deer off with a headshot is dependent on what you plan to do with it. If it's a trophy buck, and you plan to do a shoulder mount, and you're only allowed to have a rifle or slug gun, then a head shot as a finisher would kinda mess things up...