Veteran Employment Through Technology Education Courses Act or the VET TEC Act This bill directs the Department of Veterans Affairs (VA) to carry out a five-year pilot program to provide eligible veterans with the opportunity to enroll in high technology programs of education. An "eligible veteran" is one who is entitled to educational assistance under the laws administered by the VA. A "high technology program of education" is one that is offered by an entity other than an institution of higher learning, does not lead to a degree, and provides instruction in computer programming, computer software, media application, data processing, or information sciences. The VA shall enter into contracts with multiple qualified providers of such programs, under which the VA shall pay: (1) 25% of the cost of providing the program upon the enrollment of an eligible veteran, (2) 25% of such cost when the veteran completes the program, and (3) 50% of such cost upon the veteran's employment following completion. The VA shall give preference to a qualified provider that offers tuition reimbursement for any student who completes the program and does not find full-time meaningful employment within 180 days. The VA shall pay a monthly housing stipend to each eligible veteran enrolled full-time in such a program.