Reunite Every Unaccompanied Newborn Infant, Toddler and other children Expeditiously Act or the REUNITE Act This bill directs the Department of Homeland Security (DHS) and the Department of Health and Human Services (HHS) to reunite children and their parents or legal guardians who were apprehended for illegal entry into the United States. DHS and HHS shall develop procedures and programs, including procedures to (1) identify separated family members, (2) allow apprehended parents or legal guardians to have frequent no-cost contact with their separated children, and (3) coordinate with foreign consulates to locate apprehended individuals. DHS and HHS shall ensure immediate reunification of apprehended children with a parent or legal guardian, but may make exceptions for the child's safety. DHS and HHS may use DNA testing to determine family relationships, but shall first use other methods, such as official documents. DHS and HHS shall develop procedures for instances where the aliens refuse to consent to DNA testing. Collected DNA information may not be used for any purpose other than family reunification, including criminal or immigration enforcement. The bill restores the Family Case Management Program, an alternative to detention program that uses case managers to promote compliance with immigration law obligations. DHS, HHS, and the Department of Justice (DOJ) shall establish the Office for Locating and Reuniting Children with Parents. Aliens shall have access to counsel inside immigration detention facilities, and DOJ may provide counsel for immigration proceedings. DOJ shall provide counsel if necessary for children and vulnerable individuals, and children's cases shall be heard individually.