Cyber Response and Recovery Act of 2021 This bill authorizes the Department of Homeland Security (DHS) to declare a significant incident in the event of a breach of a public or private network and establishes a Cyber Response and Recovery Fund. Specifically, DHS may make the declaration upon determining that a specific significant incident has occurred or is likely to occur imminently and that otherwise available resources, other than the fund, are likely insufficient to respond to or mitigate the incident effectively. Upon a declaration, the Cybersecurity and Infrastructure Security Agency must coordinate (1) the response activities of each federal agency; and (2) with other responding entities, including local governments and law enforcement agencies. A declaration or renewal of a declaration may not include the name of any affected individual or private company. The fund shall be available for the coordination of such activities and for response and recovery support.