I'm bucking the trend here....
But I am correct.
The 9mm has the capacity to kill deer at any range you can hit them. Period.
HOWEVER, in order to make clean kills, one has to take into account the limitations of the round, and the shooter! Take, for instance the report of a deer taking several 9mm rounds through the shoulder and not only getting away, but recovering. Amazing, but I am sure quite true. Also quite non-typical. But things like this do happen.
I have personally seen a deer (and a small doe at that) take 21 (twenty-one!!!!) hits from .30 caliber rifles (.30-30, .30-06, & .308) and still have enough life left afterwards to get lost to the hunting party in a swamp. Found that dead deer the next weekend. All hits were in the "right"spots, but that deer simply didn't quit. Each shot put it down, and it came right back up after about 20-30m seconds and ran. This is one of the exceptions, and a very rare one at that, but it did happen.
The 9mm Luger out of a carbine has enough energy to make a clean kill, but bullet construction plays an important part in whether or not the bullet gets where it needs to go to make the kill. FMJ is not legal for hunting, any place I know of, and JHP/JSP 9mm rounds are designed with people in mind. And, like the .357Mag 125gr, when fired out of a carbine barrel, the extra velocity means they open up even sooner, and penetrate less. One of those 5 9mm shots to the shoulder that didn't drop the deer might have dropped it like lightning, had it hit the spine in the neck.
Bullet construction, velocity, and PLACEMENT are what matters most. Asking a bullet to break heavy bone and go on to penetrate deeply when it is not built for it, is asking for failure. Match the shots to what you are shooting, other wise, your results are going to be ....suboptimal.
Game laws are based on sportsmanship, and the general level of the hunters. You can kill deer with anything that shoots, with bows, or with pointy sticks and rocks. BUT most people hunting will not do a good job of it using the "lighter end" guns.
There isn't a deer on the earth that will not fall to a .357, or a 9mm, or even a .22LR, IF the shooter does their job of placing the shot in the right place, for the caliber (and bullet) being used. IF that means getting close, then it means getting close, or not shooting!
Many calibers are not legal for deer in many places, not because they are incapable of killing deer, but because the majority of the hunters are not capable of making clean kills with them, in the experience of various game depts.
Once upon a time, we shot and killed deer with .36 and .40 caliber muzzleloaders, firing round balls. That level of energy and penetration worked then, and will work today. The main difference is those frontier hunters, having only one shot, took great pains to ensure that when they did shoot, they were successful. Or so the stories say.
Do everything right, and you can kill deer with the 9mm carbine, and do it cleanly at ranges that would astound some people here. Do even ONE thing wrong, and you will be chasing a cripple, at best. That is why rounds like the 9mm are not considered good deer rounds, not because they can't do the job, but because when things are less than perfect, they fail more often than many better choices.
That's something important to game depts, and to ethical sportsmen as well.