Vet Center Improvement Act of 2021 This bill requires the Department of Veterans Affairs (VA) to evaluate productivity expectations for readjustment counselors of Vet Centers, including by obtaining feedback from counselors that must be audited annually for five years by the Government Accountability Office. The VA must also (1) develop and implement a staffing model for Vet Centers, (2) standardize descriptions of position responsibilities at Vet Centers, and (3) award grants to address food insecurity among veterans and family members of veterans who receive services through Vet Centers or other VA facilities.