Housing Innovation Act of 2022 This bill establishes the Office of Housing Innovation within the Department of Housing and Urban Development. The office must establish grant programs to support (1) planning efforts that address housing needs in urban areas; (2) research and studies to support, inform, and advise local governments on their plans for new housing and community development; and (3) outreach programs focused on issues related to housing, community development, and regional planning. The Government Accountability Office must report on the effectiveness of the grant programs.