Consular and Administrative Support Act of 2020 This bill requires the Department of State to take certain actions in responding to an outbreak of an infectious disease such as COVID-19 (i.e., coronavirus disease 2019), authorizes additional paid leave for certain agency employees involved in foreign affairs, and extends certain special immigrant visas. The State Department, in responding to an outbreak of an infectious disease, must establish a Repatriation Task Force. The duties of this task force shall include (1) protecting and safeguarding the needs of U.S. nationals, including repatriation and other health and medical needs; (2) managing all communication and coordination with U.S. nationals regarding their needs; and (3) supporting American Citizen Services with repatriation and other health and medical needs. Additionally, in responding to such an outbreak, the State Department must activate a single, comprehensive approach to incident management for the National Passport Center and the National Visa Center. The State Department must also extend the period of validity and the application period for certain special immigrant visas that have been affected by current travel restrictions and health concerns due to COVID-19. Further, specified independent agencies involved in the conduct of foreign affairs (e.g., the Peace Corps) may provide additional paid leave to address employee hardships resulting from COVID-19.