You can in principle try to embed an iframe in the popup to show the webapp, but a popup page is completely destroyed when you close it, so it won't stay connected that way.
You would probably need another way.
It does not seem like Hall has any public API that you can use.
So probably your only option is to use a content script to interact with Hall opened in a tab. If you're adventurous, you could try to embed an iframe in your background page and interact with it there, but Hall may disallow framing.
All that said, I'm not so sure Hall will appreciate creating alternative software to interact with it.