911 Community Crisis Responders Act of 2023 This bill authorizes grants to dispatch unarmed, specialized health care or social service providers to respond to nonviolent emergency calls instead of law enforcement officers. The Substance Abuse and Mental Health Services Administration may award grants to states, tribal governments (and consortia of tribal governments), and localities. Recipients may use the grants for hiring and training necessary staff, working with 9-1-1 dispatchers and 9-8-8 call centers to appropriately dispatch services in response to nonviolent emergency calls, and connecting individuals with local health and social service providers. (The 9-8-8 system provides callers with mental health or suicide crisis support and resources.)