You're correct in your guessing, not possible without a heck-of-a-lot of third party stuffs.
Overall, it COULD be designed by someone wanting to write some third party dlls/libraries, BUT, no developer in thier right mind should ever do this.
All it would take is another malicious person to reverse engineer how it's done, release another dll as thier own, then the whole GMod Lua world becomes virus/worm ridden because it just got access to executables on everyone's computers.
The only limited way I could see it working now was if the Gmod HTML engine allowed non-standard links (like, steam://blah/appid URLs start some multiplayer games from websites).
1, I don't know if the Source/Gmod HTML engine allows for non-standard URLs, and 2) I've no idea if TeamSpeak has such a url protocol for web pages that would start it.