Military Child Care Expansion Act of 2021 This bill modifies and implements policies to update child care facilities and administration at the Department of Defense (DOD). Specifically, the bill removes certain requirements for members of the Armed Forces to receive funding for an in-home child care provider under the DOD pilot program that provides financial assistance to such members who pay for in-home child care. The bill also allows DOD to expand the pilot program to additional locations (currently it is authorized for five). DOD must establish a 10-year pilot program to expand access to child care for members of the Armed Forces by entering into agreements with public or private child care facilities or development centers. Under the bill, DOD must determine the (1) root causes contributing to poor or failing facility conditions at its child development centers, and (2) total cost to improve the conditions of such centers. The Government Accountability Office must review the determinations and make recommendations on how to improve DOD child development center conditions. Lastly, DOD must utilize its existing authority to carry out minor military construction projects to construct child development centers.