Servicemember Apprenticeship Act This bill requires the Department of Defense (DOD) to conduct a study to identify the private entities participating in Skillbridge that offer positions in registered apprenticeship programs to members of the Armed Forces who are eligible for Skillbridge. Skillbridge provides members of the Armed Forces with opportunities to gain civilian work experience through training, apprenticeships, or internships and connects service members who are transitioning to civilian life with job experiences. DOD must consult with officials and employees of the Department of Labor who have experience with registered apprenticeship programs to facilitate DOD entering into agreements with entities that offer apprenticeship programs in areas where DOD determines few positions are available to members of the Armed Forces.