Military Housing Readiness Council Act This bill establishes the Department of Defense Military Housing Readiness Council to review and make recommendations to the Department of Defense (DOD) regarding policies for privatized military housing. The council must also monitor compliance by DOD with statutory improvements to policies for privatized military housing and make recommendations to improve collaboration, awareness, and promotion of accurate and timely information about privatized military housing, accommodations available through the Exceptional Family Member Program, and other support services among policymakers, service providers, and targeted beneficiaries. The council must make all documents publicly available until the council ceases to exist.