Global Electoral Exchange Act of 2019 This bill authorizes the Department of State to establish a Global Electoral Exchange Program to promote sound election-administration practices around the world. The State Department may award grants to qualified, tax-exempt, U.S.-based organizations that have expertise and experience in relevant topics, such as election-system integrity. Grants may be used to design programs to bring relevant individuals, such as election administrators and poll workers, together to study and discuss election procedures. The State Department shall periodically report to Congress on the program's activities.