Mental Health Access and Gun Violence Prevention Act of 2023 This bill authorizes FY2024 appropriations for the Department of Justice, the Department of Health and Human Services, and the Social Security Administration to (1) increase access to mental health care treatment and services, and (2) promote reporting of mental health information to the National Instant Criminal Background Check System. (This system is used to query federal, state, local, tribal, and territorial criminal history record information and other records to determine an individual's eligibility to receive, transfer, or possess firearms.)