Ah, That's as far as i can tell easy enough to do.
I'll add the necessary endpoint in the next release.

I do think the Core needs some modifications. As far as I remember I don't think it has a hook that fires when someone on the webadmin talks. You can however add messages by calling the WEBLOG functions.
Hmm. Seems like there's none indeed.
I'll have to make a PR, then.
Shouldn't be too hard. You can see how the commands are done. A hook would look allot like that.
Update!
I've added web and console(currently write-only) endpoints, and created a PR for Core with necessary functionality
(
https://github.com/mc-server/Core/pull/122 ).
Note: Chat coming from the web chat is actually coming from web-chat-chat due to the naming scheme for endpoint sources. Not anymore!
Any plans to support matrix?