Smithsonian Modernization Act of 2022 This bill establishes that the Smithsonian Institution must be subject to the Freedom of Information Act, the Privacy Act, and certain provisions relating to open meetings. (The Smithsonian's policy is to respond to certain requests for information consistent with these acts, even though in 1997, a federal appeals court ruled that the Smithsonian is not subject to the Privacy Act.) The bill also modifies the procedures for selecting members of the Board of Regents of the Smithsonian Institution by having the President select the members from a list of nominees submitted by Congress.