Yes, the mail is always delivered on Flag Day, as long as Flag Day does not fall on a Sunday. Flag Day is not included in the postal holiday calendar.
June 14 was designated as National Flag Day under an act of Congress signed in 1949. That date was selected because it was the anniversary of the adoption of the U.S. flag. In 1971, another act of Congress declared Memorial Day a national holiday. Today, federal law, 5 U.S.C. 6103, establishes Memorial Day and several other days as public holidays for all federal employees, but that list does not include Flag Day.