Tribal Health Data Improvement Act of 2021 This bill expands tribal access to public health care data and public health surveillance programs. It also reauthorizes through FY2026 the National Center for Health Statistics, which is part of the Centers for Disease Control and Prevention (CDC), and requires the CDC to take certain actions to address the collection and availability of health data for American Indians and Alaska Natives. Specifically, the Department of Health and Human Services must (1) establish a strategy for providing data access to Indian tribes and tribal epidemiology centers; and (2) make available all requested data related to health care and public health surveillance programs and activities to the Indian Health Service, tribes, tribal organizations, and tribal epidemiology centers. Next, the CDC must make grants to and enter into contracts with tribes, tribal organizations, and tribal epidemiology centers for data collection and related activities. Among other requirements, the CDC must (1) develop guidelines for state and local health agencies to improve birth and death record data for American Indians and Alaska Natives; (2) enter into cooperative agreements with tribes, tribal organizations, urban Indian organizations, and tribal epidemiology centers to address certain inaccuracies related to records for American Indians and Alaska Natives; and (3) encourage states to enter into data sharing agreements with tribes, tribal organizations, and tribal epidemiology centers to improve the quality and accuracy of public health data.