Veterans Serving Veterans Act of 2021 This bill updates the Department of Veterans Affairs (VA) current recruiting database by requiring the VA to maintain a single searchable database to be known as the Departments of Defense and Veterans Affairs Recruitment Database . With respect to vacant positions for critical jobs or mental health professionals, the database must list (1) the military occupational specialty or skill that corresponds to the position, and (2) each qualified member of the Armed Forces who may be recruited to fill the position before such qualified member has been discharged and released from active duty. Qualified members of the Armed Forces are those service members who (1) are being separated or recently separated from active duty, (2) elect to be listed in the database, and (3) have been determined by the VA to have a military occupational specialty. The database must list the following for each qualified member of the Armed Forces: name, contact information, expected discharge date, and military occupational specialty. The VA must implement direct hiring and appointment procedures for vacant database positions and may authorize relocation bonuses. The bill also requires the VA to train and certify veterans who served as basic health care technicians in the Armed Forces to work as VA intermediate care technicians.