This is a very complex situation and of course we have only heard part of it. I feel sad thinking of any woman of 96 wanting to see a daughter and seemingly being prevented. I don't think the sister whose home it is can be forced to allow her sister into her home but are there any other ways of helping the mother see her daughter? She may be housebound but possibly not. Family took my mil out in a wheelchair, to the coast, out for lunches, to family homes until 2 months before her death at aged 94. You anbu to want your mother to see your sibling op, but are there ways in which you could facilitate this?
The situation is complex because, although her refusal to allow her sister to visit must be very distressing to your mother, your carer sister's full time care of your mother is a massive commitment and your mother choosing to live there could indicate that they have a close relationship in other ways.
You may have already tried this, but I would approach a conversation by first acknowledging how much your sister does for your mother. You could point out that as she loves your mother very much she might well be consumed with guilt if she keeps up this refusal to allow her sibling into her home until it is too late. I would keep out of the whys and wherefores of the original row if you can but gently press the case, pointing out that letting the other sister in would both be the right thing to do for your mother and also a generous gesture on her part in the light of her own personal animosity.
Certainly if your mother dies before your sister relents, it seems very unlikely that the sisters would ever reconcile. 