Restoring Integrity to America's Elections Act This bill revises provisions regarding the Federal Election Commission (FEC), including to change FEC membership. The bill reduces the number of members of the FEC from eight to five, including by removing the Secretary of the Senate and Clerk of the House as ex officio members. No more than two members may be affiliated with the same political party. A Blue Ribbon Advisory Panel shall recommend to the President individuals for nomination to the FEC. The President shall appoint the FEC chair, subject to Senate confirmation. The bill distributes the FEC's powers between the chair and the other FEC members. The bill modifies the process for (1) the FEC to initiate an investigation, and (2) a party aggrieved by the FEC's dismissal of the party's complaint or the FEC's failure to take action on the party's complaint. Persons who submit written comments regarding requests for advisory opinions must be given an opportunity to appear at FEC hearings on such requests. The FEC's administrative penalty authority is extended permanently. FEC forms must allow for the use of an accent mark as part of a person's identification. The bill provides statutory authority for limitations on ex parte communications as applied to FEC members and employees. The bill specifies that FEC attorneys may represent the FEC before the Supreme Court.