State Opioid Response Grant Authorization Act of 2022 This bill reauthorizes through FY2027 and otherwise changes the State Opioid Response Grant program. This program, which is managed by the Substance Abuse and Mental Health Services Administration, provides funding for states, territories, and Indian tribes and tribal organizations to address the opioid crisis. Specific changes to the program include (1) expanding its scope to also address stimulant use and misuse, (2) establishing a funding methodology and minimum funding allocations, and (3) allowing the use of grant funds for recovery support services. The bill also requires the Government Accountability Office to report on issues concerning the funding and other aspects of the program.