Is it appropriate to ask a former professor to order a book for me through an inter-library loan?
I don't see an issue with this. In fact, you can probably do it yourself. Many libraries will offer library privileges to people for research. I live in a small town with a nice library. My local librarian has, on occasion, gotten such resources for me through contacts with a university library. 

Libraries generally are there to help. They need some assurance that you will treat such privileges properly, of course, but you might just go ask to see what is available to you. 

But, the professor might be willing to do it and will certainly serve as the trusted intermediary that might be needed.