The facility wants me to sign a waiver for my mother to use the equipment. Should I sign?

Without reading it, it's just a guess as to the intent, but since it's equipment you provided, the facility probably does not want to be culpable or responsible in the event of equipment failure, use, etc. that might harm your mother.

I learned at the first rehab facility with which we had experience that rehabs can't or won't accept equipment from families b/c of liability issues - they didn't purchase it, maintain it, and don't know of any defects that they might otherwise if it was their rented or owned equipment.