Responsibility for Unaccompanied Minors Act This bill amends provisions relating to responsibilities of the Department of Health and Human Services (HHS) for the care of unaccompanied alien children. The Homeland Security Act of 2002 is amended to authorize the Office of Refugee Resettlement of HHS to: (1) care for an unaccompanied alien child until the earlier of the conclusion of the child's immigration proceedings or until the child reaches 18 years of age, and (2) evaluate a child's need for follow-up services prior to placement with a sponsor. The William Wilberforce Trafficking Victims Protection Reauthorization Act of 2008 is amended to require a sponsor of an unaccompanied alien child to execute a sponsor care agreement with HHS. An agreement must provide for the child's physical and mental well-being, ensure the child's presence at immigration hearings, and ensure compliance with federal court or agency orders. HHS shall (1) reassume physical custody of a child if a non-parent or legal guardian sponsor fails to comply with a sponsorship agreement, and (2) notify state welfare agencies before releasing children to sponsors within a state. HHS shall perform background checks on potential sponsors and other adult household members before releasing a child to a sponsor. The Department of Justice shall increase the number of immigration judge teams.