Supporting Veterans in STEM Careers Act This bill addresses the participation of veterans in STEM (science, technology, engineering, and mathematics) fields, including by making veterans eligible for certain National Science Foundation (NSF) programs. (Sec. 3) The bill directs the NSF to (1) encourage veterans to study and pursue careers in STEM 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 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 improve veteran and military spouse equity and representation in STEM fields. (Sec. 4) The Government Accountability Office must study (1) the academic success rates of student veterans pursuing an undergraduate degree in STEM, computer science, medicine, or other fields identified as meeting national needs; and (2) the barriers faced by such students in pursuing such degrees.