Reunite Every Unaccompanied Newborn Infant, Toddler and other children Expeditiously Act or the REUNITE Act This bill directs the Departments of Homeland Security (DHS) and Health and Human Services (HHS), after consultation with humanitarian organizations, child welfare agencies, and states bordering Mexico, to promulgate and publish a rule describing efforts that DHS and HHS will undertake to aid apprehended parents or legal guardians in locating and reuniting with any children (under age 18 with no permanent immigration status) separated from them at or near the port of entry, or within 100 miles of the border. DHS, HHS, and the Department of Justice (DOJ) must provide apprehended parents or legal guardians separated from their children with written notice, in a language understandable to them, of any rules or guidance that may assist them in locating and reuniting with their children. The bill permits DHS and HHS to use DNA testing to determine family relationships subject to certain requirements, after using other commonly-used techniques for determining such relationships. DHS must establish secure alternative programs that incorporate case management services in each DHS field office to ensure appearances by alien parents and children at immigration proceedings and public safety and to provide for community-based alternatives to detention. DHS, DOJ, and HHS shall establish an interagency Office for Locating and Reuniting Children with Parents for facilitating the reunification of children and apprehended parents or legal guardians after entering the United States. The bill requires DOJ to appoint counsel, free of charge, for children and vulnerable aliens (e.g., persons with disabilities and victims of abuse, torture, or violence) in immigration proceedings. The bill amends the Immigration and Nationality Act (INA) to limit bonds for detained aliens to $1,500, except for aliens who have committed an aggravated felony (as defined by the INA).