I ended up writing a script like this that logs into ftpserver.com (fictional) with login / secretpassword and makes its contents available as ftpdir/. To log out, I unmount ~/.avfs.
#!/bin/sh
avfsd ~/.avfs
rm ftpdir
mount-avfs ftpdir /#ftp:login@ftpserver.com
rm ftpdir
echo secretpassword >~/.avfs/#ftp_ctl:login@ftpserver.com/password
ln -s ~/.avfs/#ftp:login@ftpserver.com ftpdir