State and Local Government Cybersecurity Act of 2021 This bill provides for collaboration between the Department of Homeland Security (DHS) and state, local, tribal, and territorial governments, as well as corporations, associations, and the general public, regarding cybersecurity. The bill expands DHS responsibilities through grants and cooperative agreements, including provision of assistance and education related to cyber threat indicators, proactive and defensive measures and cybersecurity technologies, cybersecurity risks and vulnerabilities, incident response and management, analysis, and warnings. The bill requires the National Cybersecurity and Communications Integration Center, upon request, to coordinate with entities such as the Multi-State Information Sharing and Analysis Center to engage in specified activities, including to (1) conduct exercises with state, local, tribal, or territorial government entities; (2) provide operational and technical cybersecurity training to such entities; and (3) promote cybersecurity education and awareness.