The amounts that become due to any child under this Act, not in excess of the amount specified in Section 1 of this Act, shall be payable to such child or, if such child is a minor, to the eligible veteran or serviceperson or guardian on vouchers approved by the Illinois Department of Veterans’ Affairs. The Illinois Department of Veterans’ Affairs shall adopt rules on how to render payments to eligible minor children of deceased veterans or servicepersons. The Department shall determine the eligibility of the children who make application for the benefits provided for in this Act; and satisfy itself of the attendance of such children at any such institution or school.