Yes, you could use that phrase in an email in a business context, as an acknowledgement of some information you've been given. It's slightly more formal than 'Thanks for letting me know', and the minimalist nature of the phrase makes it suitable for a brief email reply.
Note that your question should be either
Do native speakers use 'Noted with thanks' in formal emails?'
(Plural for native speakers in general)
Would a native speaker use 'Noted with thanks' in a formal email?'
(Singular for a hypothetical situation).