By professional conduct, I refer to issues related to how a researcher manages his/her research group, including funding (both research and salary) and human resource issue.
For example, there is the Office of the Independent Adjudicator in the UK that reviews student complaints against higher-education providers. But the service offered is restricted to students, and not directly relevant to the professional conduct of a researcher.
An example of such an organisation in the financial field is the Financial Conduct Authority, who oversees the professional conduct of both organisations and individuals.