Extinction Crisis Emergency Act of 2021 This bill requires the President to declare the global wildlife extinction crisis a national emergency and to take certain actions to restore wildlife. Specifically, the President shall direct the U.S. Fish and Wildlife Service and the National Marine Fisheries Service to review all species that are identified as critically imperiled, critically endangered, endangered, or vulnerable, and to use all authorities to propose and finalize protections for all species that may warrant protection; and establish new national wildlife refuges and new marine protected areas. The Departments of Commerce and the Interior must, after accepting public comment, conduct annual reviews of every nation's compliance with any international fishery conservation program or international program for endangered or threatened species, and take specified actions. The President must enlist all federal agencies to help fight the wildlife extinction crisis by directing each federal agency to take specified actions to prioritize the recovery of endangered species and to rebuild healthy wildlife and plant populations, including by fully integrating climate change concerns into the conservation and recovery of endangered species. The President must direct the Department of Defense to prioritize the protection of endangered species and rare habitats on U.S. military installations around the world.