I'm reading the documentation for WWW::Scripter::Plugin::Javascript and it's not immediately clear to me how to use it to access the HTML that is released by the html.js script that runs on the webpage. Do I somehow use WWW::Scripter::Plugin::Javascript to force the script to run on the website and then capture the output?
No. The plugin adds support to the browser (WWW::Scripter) so it can execute javascript on webpages -- sometimes it works, sometimes it doesn't, its alpha-quality
It seems you have a conceptual gap in your knowledge, see learn about the internet,Web Programming: For Beginners, to get an overall picture of how the internet works, how tcp/ip, sockets, websites, html, ajax, all fit together.