I'm setting up X with TWM so that folks here can access one specific graphical application. I have it setup so one can VNC in, log-in using XDM and have said application automatically start up.
What I'd like to do is have it automatically log-out of the session once that application closes. Is there a trick I can do in the xinit script, or find some way to make TWM dependent on this particular application?
An X session ends when its “main” client process ends. Often that's the window manager; in setups it's a separate session manager process. In your setup, just kill the twm process when the application exits.
If your .xinitrc runs twm in the background then you can do
twm &
myapplication
kill $!
If you make twm start your application with f.exec then use f.exec "myapplication; kill $PPID" instead of f.exec "myapplication".
If you only ever want to display a single application and it only displays a single window then you could just start it full screen instead of the window manager. A window manager isn't much use when there's a single application. This is known as a “kiosk” configuration.