Is it appropriate to ask a former professor to order a book for me through an inter-library loan? I am not a member of an academic institution right now, but am working on a paper for publication, partly to improve my prospects for admission to PhD programs after a long gap in my education. Two absolutely necessary books for my research are only available in academic libraries (one is also for sale on AbeBooks for about $450).

So my question is, would it be appropriate to email a former professor and ask him to get these books through an inter-library loan (ILL) for me so that I could skim them for several hours, take necessary notes, and return them on the same day?

This professor has taught me in the same field, and was personally supportive. Aside from former professors, I don't know anyone else in an academic institution, and am not in a position to fly across the country to visit libraries where these items are held.
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.