National Climate Bank Act This bill establishes and capitalizes a National Climate Bank. The independent, nonprofit bank must invest in clean energy technologies and infrastructure to reduce greenhouse gas emissions. The national bank's investments and procurements division must seek to facilitate affordable investment and procurement, including in low-income communities and communities of color, in key project areas (e.g., renewable energy or climate resiliency measures). The national bank's start-up division must support the creation of new green banks by states or other political subdivisions. The new banks must be public or nonprofit specialized finance entities that use finance tools to mitigate climate change. The national bank may provide financing for such entities. In addition, the bank must explore the establishment of a cash for carbon program to remove greenhouse gas emissions from the power system. The program may use market mechanisms to expedite the retirement of carbon-intensive power generation facilities (e.g., coal-fired power generation facilities), acquire carbon assets for the purpose of reducing emissions, and invest in communities negatively affected by the loss of those facilities or assets.