Farm Board Act of 2023 This bill further specifies the active producers that must be included as members on the Federal Crop Insurance Corporation (FCIC) Board of Directors. FCIC is a government corporation that administers the federal crop insurance program. Of the four active producers who must be included on the FCIC Board, the bill requires at least one specialty crop producer, one producer that actively engages in livestock production and crop production of agricultural commodities, and one underserved producer. (Current law requires that, of the four active producers on the board, at least one must be a specialty crop producer.) The bill applies to appointments made on or after the date of the bill's enactment for either (1) service beginning on or after October 1, 2024; or (2) the purpose of the October 1, 2024, FCIC Board election (or a subsequent FCIC Board election).