My wifes mother is in nursing home in MA. She's been harrassed and wants a new emergency placement ASAP. How do we make this happen?

Your question says NH? If of sound mind why is she is a NH.? Does she need 24/7 care? Usually Healthcare proxy doesn't come into effect if the person can make there own decisions. Who is paying for her care? If Medicaid its going to be hard to change over, not impossible, but hard. With the COVID too. Then its if there is a bed available, especially with Medicaid.

Just curious, what kind of harassment?

Well, this would be much like a child saying they had been harassed at school and want an immediate transfer. Wouldn't be easy. I think calling in the ombudsman is a very good idea. I think also, given that people who are in Assisted Living are often dealing with somewhat altered minds, the things that happen and are said can be quite odd. I told my bro that he and his gang were like a bunch of hippies in one of those 60s communes, that required periodic community meetings to work stuff out.
I am afraid our "wants" become almost impossible to fulfill at times.
More details on the type of harassment might change my mind, but I think I would try to look at this as a misunderstanding that should be worked out.

In what way is she being harassed?
Would a phone call to the State Ombudsman help the situation?
If she is of sound mind / not been declared incompetent she can arrange transfer to another facility.