Taiwan Fellowship Act This bill directs the Department of State to establish a program to provide two-year fellowships in Taiwan to qualifying U.S. government employees. Each program fellow shall work in (1) a Taiwanese government agency; or (2) a nongovernmental organization, such as the American Institute in Taiwan, whose interests are associated with the interests of the fellow's employing U.S. government agency. (The United States does not have official relations with Taiwan as a result of China's position that Taiwan is part of China. The American Institute in Taiwan is a private organization that has a contract with the State Department to represent U.S. interests in Taiwan and provide consular services.) The State Department and the American Institute in Taiwan shall select a nonprofit organization to implement the fellowship program.