Building More Housing for Servicemembers Act This bill addresses the availability and quality of housing for members of the Armed Forces. First, the bill requires the Department of Defense (DOD) to report to Congress on the housing shortage for members of the Armed Forces. DOD must prescribe guidance for eligible entities and landlords regarding acceptable housing standards for privatized military housing. Under the bill, an eligible entity is any private person, corporation, firm, partnership, company, or state or local government that is prepared to enter into a contract for the construction of housing units and ancillary supporting facilities. DOD must establish a five-year pilot program to assess the feasibility of using the rental partnership programs of the Armed Forces to assure tenants for eligible entities to secure financing to construct privatized military housing. Additionally, DOD must coordinate with the Department of Housing and Urban Development (HUD) to develop a five-year pilot program to provide financial incentives to eligible entities to build privatized military housing, or to purchase or lease existing facilities, to house members of the Armed Forces and their dependents and low-income individuals and families. Finally, DOD and HUD may jointly operate a grant program through the Office of Local Defense Community Cooperation within DOD to build housing for members of the Armed Forces and their dependents, and low-income individuals and families. Household income limits for eligible entities to receive a grant must not differ based on whether a household includes a member of the Armed Forces.