Restoring Confidence in America's Elections Act This bill amends the Help America Vote Act of 2002 to: (1) prohibit the purchase or use in a federal election of an electronic voting system that does not produce a voter-verified paper record; (2) prohibit the imposition of conditions on voting by mail; (3) make mandatory the availability of early voting; and (4) require sufficient polling places, equipment, and resources to facilitate voting. The National Science Foundation shall make grants to: (1) study, test, and develop accessible ballot voting, verification, and casting mechanisms and devices and best practices to enhance the accessibility of paper ballot voting for individuals with disabilities, for voters whose primary language is not English, and for voters with difficulties in literacy; and (2) research the development of election-dedicated voting system software. Each state shall, except when the winning candidate had no opponent or received 80% or more of the votes, manually audit federal election results, without advance notice to the precincts selected. The bill amends the Federal Election Campaign Act of 1971 to make it unlawful for a chief state election administration official, unless such official or an immediate family member is a candidate, to take an active part in political management or in a political campaign for federal office over which such official has supervisory authority. The Department of Justice must initiate an investigation of an allegation of voter intimidation or suppression within 30 days of receiving such an allegation. The Election Assistance Commission shall award grants for the development of voting systems that produce a voter-verified paper record of each vote cast, are compatible with commodity accessibility devices, and are fully accessible for the use of individuals with disabilities.