You gotta start with the round being used. Here is why: the round determines 2 things, the size of the mechanism, and the amount of recoil.
Lets talk terms for a second. To avoid a little bit of pedantry lets get a little more specific than 'Machine Gun'. The weapon in question for Sister Superior is going to be a Detachable Box Magazine fed Fully Automatic weapon. That means that the rounds are loaded in to a metal box with a feeding mechanism. the Firing mechanism will use either the recoil of the round being fired or the expansion of gasses from firing to remove the current round's brass, chamber the next round, and fire again as long as the trigger mechanism is activated.
Now that we have disposed of that little bit of trivia, lets talk about recoil. This is pretty much Newtons Third Law in Action. To oversimplify the math, The Mass of the slug times the velocity of the slug leaving the barrel is going to have an equal and opposite reaction. The mass of the weapon (usually many many times the mass of the slug) travels in the opposite direction and will impact over a much greater surface area (usually the butt of the rifle) repeated each time the thing fires. The felt recoil is what is left over after whatever energy is dissipated in the mechanism of the weapon. Remember you have a tiny chunk of lead going one way, and a large chunk of other metal going the other, spreading the impact over a greater surface area. This is why firing powerful rounds can bruise the heck out of your shoulder. Given that this weapon will be impacting a surface area that is pretty small (the size of the cross section of her normal arm's stump) you want to fire something that has less recoil. 9mm is a common round for many compact submachine guns like Uzis and Mac 10's. They are hard to control when firing on fully automatic, because the mass of the weapon is low. The recoil comes back along the mechanism and tries to 'lift the barrel because the wrist becomes a pivot point. So I'm thinking that Sister Superior's arm gun may not have exactly the same problem, but it's not a trivial thing.
Next you have the problem of the mechanism's size. It has to be small, otherwise Sister Superior will have to change her name to Sister Popeye. The Mechanism of a 9mm fully automatic 'might' be made small enough, Anything bigger goes right out the door due to recoil problems. The mechanism for something smaller might be better. I would think maybe a .22 Mag would be almost Ideal for many reasons.
First, .22 Mag rounds are small. You can fit a lot of them in a small magazine. Look at this weapon here. It is roughly the same size as a mid size 9mm pistol. The difference is that it holds 30 rounds in the magazine where a similar sized 9mm would hold maybe 9 rounds. 9 rounds, reload, 9 more reload would be silly. 100 round magazines for 9mm exist, but when loaded, they are pretty heavy. That's going to make it harder to maintain her point of aim. .22 Mag rounds will be much much lighter. Recoil from the .22 Mag is pretty low. It's enough to eject the brass and load the next round, but you don't need huge, heavy components to do so. The .22 Mag can also be lethal at close range. She is not looking to pick someone off at 500 meters with this thing.
I can see Her weapon being chambered in .22 Mag. She can carry a lot of Box Magazines filled with 30 rounds each, and never disturb the lines of her Habit. The mechanism would be small enough to fit in her 'forearm'. The recoil will also not pound her stump to jelly or knock her over. She'll be able to aim without problems too.