Restoring Integrity to America's Elections Act This bill revises provisions regarding the Federal Election Commission (FEC), including to change FEC membership. Specifically, the bill reduces the number of appointed members of the FEC from six to five and permits no more than two members to be affiliated with the same political party. Further, it removes the Secretary of the Senate and Clerk of the House as ex officio members. Next, the bill establishes the Blue Ribbon Advisory Panel to 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. Further, it requires the FEC to ensure that its forms allow for the use of an accent mark as part of an individual's identification. 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 to seek judicial review in federal court. Additionally, individuals who submit written comments regarding requests for advisory opinions must be given an opportunity to appear at FEC hearings on those requests. The bill also permanently extends the FEC's administrative penalty authority. Finally, the bill provides statutory authority for limitations on ex parte communications as applied to FEC members and employees. It also specifies that FEC attorneys may represent the FEC before the Supreme Court.