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There is no traditional (non compound) bow with which somebody with shoulder problems could kill deer. A compound bow set at 45 - 55 lbs could kill deer if your friend could pull it.

I hunted with a bow for years and saw many deer killed with bows. A 40 lb recurve will kill a deer quite handily if you keep the shots at 40 yards or less or even up to 50, the problem is the trajectory not the penetration power when it comes to bows with a light pull. If the guy can pull a 40 lb recurve he can kill deer quite handily as long as he learns to shoot. I knew one guy who hunted with a 38 lb recurve and killed 10 bucks total in Ohio and CA with it. Never lost one, was always able to find them dead.

The big advantage of a heavy drawing bow, especially compounds, is the trajectory, which makes it much easier to hit deer.

I have another friend that bow hunts from tree stands, and he tells me to wait after shooting the animal for about 15 to 20 minutes before you track the animal down.
He said that if you go after the animal immediately, it'll keep running way off and slowly bleed out,
where if you wait, the deer will go a short way, then lay down, thinking it's hid, and bleed out there.
But I don't know. That's just what he said.