....The problem is my daughter was born without a bone in her left arm limiting movement.
.....When she was younger she had more range of motion but now that shes 11, it's become so difficult that she cannot support a rifle with non-dominate hand....
I'm an Adapted PE specialist. I have a number of students that need modified baseball bats, hockey sticks, tennis racquets, etc.
I'm unclear on a couple of things:
Is her inability to support a rifle due to weak shoulder strength or weak hand/grip strength?
If she has sufficient strength, is it more a matter of an efficient grip on the forearm?
I know of one adult who shoots bullpups because he lost his entire arm. It puts the balance closer to his body. In addition to factory bullpup designs, there are several aftermarket bullpup stocks for shotguns, Ruger 10/22's, AR's and AK's.
.... She cannot turn her palm facing upward counter-clockwise......
With left hand, that's pretty difficult for anyone. If you extend your left arm forward with palm up, you aren't likely to get much range of motion past horizontal.
....She also cannot use a vertical style fore-grip because the angle is just a little to extreme for her limited range of motion.
From your comments regarding Sten guns, I presume a fore grip mounted at a right angle on the left side of the rifle would work?
Here is an interesting grip that may work:
http://www.thetruthaboutguns.com/2017/10/jon-wayne-taylor/ryker-usa-fist-grip/
....We've tried some handguns from time to time but being that her hands are still very small and she can't legally own one till 21 I'm dismissing that option for the time being.
Is that a state law where you live? Because Federal law most certainly allows someone under age 21 to use or possess a handgun.....only prohibiting them from purchasing through a licensed dealer.
There are a number of handguns that fit even the smallest hands. I have several customers who have seven year old daughters shooting .22's. (Walthers & Rugers)