9X19 is a pistol round. That's it. No magical bullet weight/powder burn barrel length going to change that. Leave it at home and use a big-game round's for big-game. If we take the 9mm case, design a rifle around it to take 65,000 chamber pressure and load the most pointed bullet you could find, it's still just too small, and it's still a limp-#### round for Big game.
Now if you are carrying one for self-defense,...... such as in the case of a Wildlife Officer,.. and you were suddenly and ferociously attacked by a big Whitetail buck who refused to comply with your instructions during a routine trailside check,....then by all means its self-defense and in such occurances, any gun is better than no gun at all.

Seriously,...there are many fine choices for novice, smaller statured shooters in lightweight fast handling carbines. From .243 Winchester through the parent .308Win is one route, even the 7.62X39 in a light rifle, an SKS perhaps would make a great start, the other route could be short, lightweight lever guns in .30-30 or 44Mag to name a few. I chose a Marlin 336TS for my wife in .30-30 with a 18.5" barrel and factory 170grains it was pleasant to shoot, but still marginal on deer if recovering one within sight is a priority, a fact I proved several times for myself. The .30-30 by the way, is a powerhouse compared to the 9mmPara ok.
It's great to get our wife's, girlfriends and kids into the sport, because firearms in general and hunting in particular need a big injection of new devotee's, but at the same time,.. we don't want to produce wounded and lost deer for their sake merely based on recoil sensitivity. They need to learn shoot a "deer" rifle properly. Hearing protection on the range is 75% of the way there, and when firing at actual game in the field we all know recoil doesn't matter one GD bit, whatever the caliber. I wouldn't let my wife fire at a moose with an M-1 carbine because she likes the recoil, no more than you should let yours fire at a whitetail buck with a 9mm Pistol.