Reading this has made me realize that I am somewhat possesive of my office chair. I'm pretty sure no one else wants it, as it's older and I've made a mess of the arm rests bashing them into my desk, but I would be quite miffed if I came to work and found out it had been reassigned. When the company was ordering new chairs, I had the chance to 'upgrade', but I didn't want to. My chair suits me perfectly, has features I like, and I see no reason to change it.
So I understand how the OP's co-worker could be upset by this. It's not so much the chair, as the way it was done, arbitrarily and behind her back, essentially. Or it could be that she is used to her chair and finds the substitute less comfortable or less ergonomically suited to her and is cranky about that. I don't think she's handling it well, and I think redirecting her to the boss, as others have suggested it about all you can do.
I also wonder if instead of ordering a new chair for her, the new chair would be for you, and you could ask her to agree to allow you to use her chair until the new chair arrives, and then promise to return her chair. She may feel better if she has some say in the situation.