National Capital Region Leadership in Environmental and Energy Stewardship Commission Act - Establishes the National Capital Region Leadership in Environmental and Energy Stewardship Commission, requiring it to: (1) analyze the overall facility environmental and energy footprint and impact of the federal government in the National Capital Region; (2) hold a nationwide competition for a cash prize to find innovative solutions to significantly eliminate such emissions and reduce dependence on the national power grid by making local combustion facilities climate neutral or implementing other technologies to achieve the same goals; (3) analyze existing and upcoming technologies which the government can implement to eliminate facility pollution and reduce dependence on the national power grid; and (4) recommend appropriate ways for the government to significantly eliminate facility pollution emissions, including greenhouse gases, and reduce its dependence on the national electrical grid.