Education Savings Accounts for Military Families Act of 2019 This bill directs the Department of Education (ED) to establish a program to provide children with parents on active duty in the uniformed services with funds to pay educational expenses. Specifically, ED shall establish a tax-exempt Military Education Savings Account for dependent children of parents in the uniformed services for the payment of the children's educational expenses. Funds in the savings account may be used for, among other things, the cost of attendance at a private elementary or secondary school or institution of higher education, private tutoring, or costs associated with an apprenticeship or other vocational training program.