It’s been a while since I last used TeamSpeak 3 (all the kids are on Discord nowadays) but the system was pretty rough in terms of integrations. I don’t think they support any kind of outside SSO. And a quick web search also gave me no results.
Yes I know Discord is the thing right now, but still there are some features missing which let teamspeak still be the point for a huge user base. I know, like I said above, there are kinda php or also ajax frameworks existing to handle such things. I saw it myself when actively playing Eve Online in some big alliances, they used such stuff. I found the following article, and I think thats pretty much that what I am looking for:
Nope, at this point TeamSpeak is a bit of a dinosaur. You can see from searching here on meta that yours is the only post on the topic.
There might be enough there for you to achieve this by writing your own plugin, alternatively if you aren’t comfortable with code but have a budget I would suggest posting on the marketplace.
If neither of those things are an option for you then there are several Discord integration options already available