National Community Violence De-Escalation Training Act of 2021 This bill establishes and expands grant programs for training on de-escalating violence and crisis situations involving individuals with behavioral health issues. Specifically, the Substance Abuse and Mental Health Services Administration (SAMHSA) must award grants to state and local governments, educational institutions, nonprofits, faith institutions, or small businesses for community-based training to de-escalate and avoid violence. As part of this grant program, SAMHSA must also establish an advisory board and develop relevant training standards. The training funded by the grants must apply evidence-based de-escalation and avoidance techniques to potentially violent scenarios that are regularly encountered by community members, including public safety officers, public transit operators, educators, and faith leaders. Additionally, the training must address implicit bias, crisis interventions for those experiencing mental health or substance use issues, and other specified topics. The bill also expands and extends through FY2027 mental health awareness training grants.