She was never considered a tenant but was informed in August by her son in law that she must be out by December 1. She is 94 years old and he has been doing what he can but she retaliates constantly by yelling at him due to the upcoming sale of the property. I am also worried that if she does not cooperate with the application for nursing home, and is not out, who will be responsible for her. She has always given both her daughter in laws a hard time and has been down right nasty with us on various occasions and now she is angry that neither of us will take her in. My sister in law is 76 and caring for her husband who has just had his third vascular by pass surgery. Please let me stress, her moving in with either of son's is not an option. I am just worried as to who is responsible for her and how will my brother in law get her out of the house once it is sold. This is a horrible situation and I just want it to go away.
So there is no doubt she will have to leave.
Homeowners have the right to say who will live in their homes. Her sons say she cannot live in their homes. So she can't. My advice is to not allow her to move in even "temporarily" until a bed opens up or she finds an apartment or for any reason, or you'll be facing this situation all over again.
So she can't stay where she is and she can't move in with a son. Now what? It sounds like you have located a suitable option for her but it is likely that you can't force her to accept it.
You don't mention what MIL's cognitive abilities are. Assuming she is still rational, explain that she MUST find a new place to live. Offer to help her with that task or ask if she would prefer having help from an agency, such as social services or adult protective services. (Don't make working with an agency sound like a threat. It is just one of her options at this point.)This must be stated calmly and politely and repeated as necessary, no matter how much yelling she does.
If it is clear after several attempts that she is not going to cooperate, then do bring in an agency. I suggest APS, but others may have additional suggestions for you.