Veterans Early Treatment for Chronic Ailment Resurgence through Examinations Act of 2021 or the VET CARE Act of 2021 This bill addresses dental care programs for veterans. Specifically, the bill authorizes the Department of Veterans Affairs (VA) to furnish administrative support to a dental care provider that provides services to a veteran that would not be covered by VA benefits. The bill also requires the VA to implement a pilot program to provide free outpatient dental services and treatment, and related dental appliances, to eligible veterans. Eligible veterans are those who are (1) enrolled in the VA health care system but are not eligible for VA dental care, (2) not receiving regular periodontal care, (3) between 40 and 70 years old, and (4) diagnosed with type 2 diabetes.