As far as I know, there is no Webmin module for supervisord. Webmin's Bootup and Shutdown module does support systemd and upstart, in addition to traditional iniscripts. It wouldn't be extremely difficult to add support for supervisord, if you wanted to go that route (and, since the module already supports to many different init systems, it is already pretty modular and easy to add stuff to). systemd has supervisor type features, and I believe upstart does, too. Since you're on Ubuntu, you probably have upstart as the standard init (newer Ubuntu versions will switch to systemd).
The documentation for that module is here, though it doesn't currently cover Upstart or systemd: http://doxfer.webmin.com/Webmin/Bootup_and_Shutdown
I'll try to add systemd docs soon.
If you wanted to tackle adding support, development docs are here: http://doxfer.webmin.com/Webmin/Main_Page#Developer_Docs
If there are features of systemd or upstart that aren't supported that you need, you can make a feature request in the github for Webmin or on the Virtualmin ticket tracker.
Also, there is a System and Server Status module, which provides similar features within Webmin; it can watch for processes, and react when something changes, like a process dying. It can restart the process, notify you, etc. Documentation for that is here: http://doxfer.webmin.com/Webmin/System_and_Server_Status