Is it an invasion of privacy to ask a lecturer for details on their LaTeX editor setup?
I don't think anybody would look on such a request as an "invasion of privacy"; if anything, most people are happy to share the names of tools they use. I wouldn't worry about it, but don't be alarmed if you don't get a response, either, since this would be a "low-priority" email for most people to respond to (unless it's someone they're actively supervising or collaborating with!).