Veteran Employment and Child Care Access Act of 2019 This bill requires the Department of Veterans Affairs (VA) to provide child care assistance to an eligible veteran for any period that the veteran (1) receives VA training or vocational rehabilitation, and (2) is required to travel to and from a facility for the receipt of such training or rehabilitation. Child care assistance may only be provided once for a child and for a period shorter than six months, but the VA is authorized to waive such limitations. The term eligible veteran is defined as a veteran who is the primary caretaker of a child and (1) who is receiving VA training or vocational rehabilitation on a full-time basis, or needs and would receive such training or rehabilitation from the VA but for the lack of child care; and (2) whose family's adjusted gross income meets certain criteria. The bill provides for various types of child care assistance, as the VA considers appropriate.