universal-pokemon-randomizer | Public repository of source code | Generator Utils library

Browsers can run a very limited set of language runtimes; essentially only Javascript and things that can be recompiled into Javascript. There's not a generic way to take a running desktop application and republish it into a browser.

Docker here is just an intermediate layer. It's very good for things like HTTP servers that are self-contained except for a network port; okay for command-line applications (if you don't mind running it as root with a command line three times as long, it works fine); and bad for GUI applications (works only on Linux, the command lines are long and arcane, and things are still highly likely to look different). It doesn't provide any magic to cross these layers.

You might be able to find some existing software that can fill some of these gaps; perhaps a browser-based X server, or maybe one of the full-blown VM systems can display a VM's desktop in a browser, or maybe you might be able to use the all-but-dead Java applet system, or maybe there's a Java layer that provides a websocket-based client and lets you run your Java-native GUI application with minor modifications.

I'd encourage you to first make your application work, and solve the "how do I actually make it appear the way I want" problem, and only then bring in Docker if it's appropriate. The layer of separation it provides can make many things more difficult especially while you're actively developing or exploring an unknown space.