Does anyone know if optometrists are able and willing to perform an eye exam and write a prescription for new eyeglasses in a long-term care facility?

Sat1209, it does depend on the facility. Most facilities I have worked at do have an optometrist that visits usually monthly. They are out there. As others have suggested, ask the social worker or DON.

At my mom's nh, there is an optometry service that visits every few months to do eye exams. Ask the DON or SW if this is available on site.

Discuss with DON or Social worker. If Mom can be put in a wheel chair and there is a van available that trnsports wheelchairs i see no reason she can't be taken to the office.
If this can't be done I don't see why an optometrist can't come to the NH and perform enough of an exam to provide new glasses. Talk to all the Eye Drs and optometrists in your area and see what ideas they have. I am sure this is not the first time this has happened.
You can buy readers of any strength on line. You choose the strength depending on how small the print is that you can read. NYS only allows OTC glasses up to 2.5 to be sold in stores but I believe PA allows the sale up to at least 4. You could buy some in a store to try on and return them if they don't work as a last resort. Even the $1 stores have them.

That would be difficult to do, considering the equipment that they use. I don't know if there are any optometrists who prescribe glasses the old-fashioned way anymore. The people at the facility would have the best idea about that. They may know of someone locally who is willing to do it. Good luck!