Supporting Veterans in STEM Careers Act This bill directs the National Science Foundation (NSF) to: (1) encourage veterans to study and pursue careers in STEM (science, technology, engineering, and mathematics) and computer science in coordination with other federal agencies that serve veterans, and (2) submit a plan to Congress for enhancing veterans outreach. The National Science Board shall provide in its annual report on the state of science and engineering in the United States relevant data on veterans in science and engineering careers or education programs. The bill provides for veterans' participation and outreach in: (1) the Robert Noyce Teacher Scholarship program to recruit and train mathematics and science teachers, (2) NSF fellowships and masters fellowships for mathematics and science teachers, (3) computer and network security capacity building grants, and (4) traineeship grants leading to a doctorate degree in computer and network security research. The Office of Science and Technology Policy shall establish an interagency working group to coordinate federal programs and policies for transitioning and training veterans and military spouses for STEM careers.