Merit Systems Protection Board Reauthorization Act of 2021 This bill reauthorizes the Merit Systems Protection Board through FY2026 and modifies its authority. Specifically, the bill requires the board to establish and collect a filing fee to be paid by any individual filing a claim or appeal. The bill requires the board to make a decision in a matter after receiving the written representations of the parties and after an opportunity for a hearing, and the bill authorizes the board to grant a motion for summary judgment. With respect to an appeal from a performance or conduct-based adverse action, the bill prohibits the board from mitigating the personnel action unless it is so disproportionate as to be without justification. Further, the bill requires that all decisions of the agency be supported by substantial evidence. In addition, the bill limits furlough appeals and provides for the reappointment of board members (currently, board members may not be reappointed). The bill also makes the board the arbiter of appeals involving the removal, demotion, or suspension of an employee of the Department of Veterans Affairs.