Export Chat history (posts) from Teams

Not really, the only thing that comes close is running eDiscovery, but that requires admin privileges and has its own issues. It's a common request though, so hopefully it will eventually be addressed.

 

Now, if you are talking about the more general case of doing this as a partner/ISV, take a look at the Graph API: https://docs.microsoft.com/en-us/graph/api/resources/conversation?view=graph-rest-1.0