fanficdownloader. I haven't tried the webapp version of it, but I'm happy with the CLI.
Many thanks for the suggestion! I've started trying it out, and though it doesn't seem to work perfectly for fimfiction.net (half the .mobi files I create from fanfics there get rejected for some reason when I email them to my kindle), it so far seems to work fine with fanfiction.net at least.
An excuse for me to learn Python so I can fix whatever it's doing wrong. :-)
EDIT: On second thought, fimfiction.net allows me to get html downloads of the stories, which I can then email to kindle anyway -- so as long as fanficdownloader works with fanfiction.net, I'm all set :-) Thanks again.