Not sure how long you're going for, but a quick check shows that a one-week wheelchair rental might run you about $70. As Chris says, the airlines will provide a wheelchair and a person who will accompany the person in the chair (and push... but that might be what I experienced because I was travelling a single mom these trips) ... note: call ahead to arrange for this for both legs of your flight ahead of time. If you're going to bring a chair with you, also notify the airlines about it.

Also, note: if you're using DME, you might have a problem with getting a chair to meet you at the bus on the way home (this was an unfortunate experience I had during a post-surgery trip and having to walk a long distance had negative consequences). Hopefully, this policy will have changed by now.

My suggestion is that you consider purchasing a chair, depending on the length of your trip. You can find them at a pharmacy, big box discounters or even Amazon (or try at Salvation Army). They're also available for rental at medical supply companies.

One last thing: there's a difference between a transport chair and regular chair. If your MIL is going to be sitting in it all day, a regular chair will be more comfortable for her.