Nonpartisan Postmaster General Act of 2021 This bill restricts the political activities of the members of the Postal Service Board of Governors, including the Postmaster General and the Deputy Postmaster General. Specifically, the bill prohibits (1) the Postmaster General and the Deputy Postmaster General from taking an active part in political management or political campaigns; and (2) the Board of Governors, the Postmaster General, and the Deputy Postmaster General from holding any political position while in office or soliciting funds for the campaign of a candidate for elected office or for a political party. An individual must make certain financial disclosures prior to assuming the duties of Postmaster General or Deputy Postmaster General.