My first thought was that cats are very adaptable. They will find a way to get around, even if they are severely handicapped.
You’re an angel for saving this one, and she will grow and learn to move and to use the box. I bet she is a delightful pet for you for many years, assuming her missing her paws is her only physical problem.
May God bless her and you and your family.
For the litter problem, see if you can get a doggie litter pan. I have two. I got them for my elderly cats when they had troubles getting in and out of the litter box. They are kind of big..One I have has low sides and a wide opening and makes a great bed for an elderly cat. The other has higher sides and a shorter but low opening that I used for the litter.
As a side note, I just adopted two kittens, one was injured. Some jerk drove by Animal Services and from a speeding truck threw a box onto the property. Someone stopped and thankfully checked it out to find 4 tiny kittens. The litter was brought to the Pet ER and one kittend did not survive, the others were ok but one had damage to the ligaments to the left rear leg..All kittens are fine but with my tiny injured one, we have some litter problems too...she is a lot smaller than her sister..Last Fri Xena The Warrior Princess was 4 lbs and her sister little Gabrielle was 2.8..and Gabrielle will leave littel “ tootsie rolls” now and then as she goes...I have yet to figure this out...she eats like a truck driver yet is so small..has been to the vet and tested for everything and is thriving but this little poopy balls...gotta stop..so anyone got ideas about that to piggy back on this general topic?
I also had a 3 legged black cat, Mocha ( meaning one without a limb in Spanish) and she got around just fine and lived to be 25.