I can't find any reference to an API that enables Rest API clients to export an existing Google Sheet to a csv file.
https://developers.google.com/sheets/
I believe there should be a way to export them.
You can use the Drive API to do this today -- see https://developers.google.com/drive/v3/web/manage-downloads#downloading_google_documents, however that will limit you to the first sheet of the document. The Sheets API doesn't expose exporting as CSV today, but may offer it in the future.