Alas there's no formal save-page-now API, but if you experiment with using it from a browser, it's not hard to call from a program: fetch https://web.archive.org/save/<url>. The return is html, but if you examine the headers you get back, you'll see that Content-Location: tells you the permanent archived name of that particular capture.
I call APIs like this "accidental APIs"! From looking at our traffic, we have quite a few programmatic users of it.
