E-bonding for Immigration Integrity Act of 2019 This bill requires an alien who is in one of the following nonimmigrant categories to post a bond prior to being issued a U.S. entry visa: (1) an alien who is from a country with a visa overstay rate greater than 1.5%; or (2) a B-visa tourist or business entrant, an F-visa foreign student, an H-visa temporary specialty profession or nonagricultural worker, or a K-visa fiance/fiancee. The Department of Homeland Security (DHS) shall (1) establish a $2,500-$10,000 bond amount, which shall be reviewed annually, for each visa category based upon overstay risk; and (2) identify nonimmigrant visa category overstay rates, including categories with overstay rates greater than 1.5%. The bond shall be released to the posting alien upon (1) receiving notification that the alien has left the United States and returned to the country of origin, or (2) the alien changing or adjusting to an immigration status for which no bond is required. The bill establishes the E-bond Enforcement Fund which shall be funded by forfeited bonds and used to ensure compliance with this bill and to administer enforcement programs. DHS shall report to Congress annually regarding: (1) visa overstay rates by nonimmigrant category, (2) categories with overstay rates greater than 1.5%, (3) bond amounts, and (4) fund information.