National Jazz Preservation, Education, and Promulgation Act of 2021 This bill establishes a National Jazz Preservation Program and a National Jazz Appreciation Program, to be carried out by the Smithsonian Institution through the National Museum of American History, to (1) preserve knowledge and promote education about jazz, and (2) further the appreciation of jazz music throughout the nation. The Smithsonian Institution shall (1) record interviews with leading jazz artists; (2) acquire, preserve, and interpret jazz artifacts; (3) continue to recognize Jazz Appreciation Month; (4) establish collaborative agreements for the sharing of such artifacts; and (5) encourage, consult with, and engage in capacity building with community-based and regional organizations with the potential to establish jazz archival collections. The bill amends the Elementary and Secondary Education Act of 1965 to allow the use of certain funds for programs to promote jazz education, which may include (1) a Jazz Artists in the Schools program, (2) a program for the development and distribution of educational materials about jazz and teacher training on jazz education, and (3) an Ambassadors of Jazz program. The Smithsonian Institution shall establish a series of jazz performances at Smithsonian affiliates that provides broad geographic access to jazz and supports public appreciation for the diversity of jazz music.