How do I find a facility right away due to caretaker's failing health?

You begin by calling facilities in your area and setting up a tour. Some may be virtual right now, others may allow an in person tour.
You can not get someone in overnight the person that will become a resident needs to have a Physical in some areas a TB skin test or Blood Test as well as be up to date on vaccinations. They will let you know what is needed. Contact the doctor right away and see how quickly they can see you (Or I guess this is for your mom) You could even ask at the Facility if the exam can be done at an Urgent Care Walk in site rather than the doctors office. As long as you have access to the immunization information I do not see why they would not permit that.

Call the patient's doctor and ask for help. Both you and the other person's doctor.