Vet Center Outreach Act of 2021 This bill requires the Department of Veterans Affairs (VA) to take specified actions to provide information on Vet Centers to service members who are transitioning to civilian life. Vet Centers are community-based counseling centers that provide social and psychological services to eligible veterans, active duty service members, and their families. Specifically, the VA must electronically transmit, within seven days after a service member separates from the Armed Forces, specified personnel information (e.g., contact information) to the Vet Center nearest to where that service member intends to permanently reside after their separation. The VA must ensure that the information is used to contact members and former members of the Armed Forces who are transitioning to civilian life within 14 days of receiving the information. As part of the Transition Assistance Program, the VA must provide members of the Armed Forces with information on how to locate Vet Centers and how to use services provided through Vet Centers (e.g., mental health services).