Liver Illness Visibility, Education, and Research Act of 2021 This bill expands and coordinates activities related to the research of liver cancer and liver disease, including causes, prevention, detection, and treatment. Specifically, the National Cancer Institute must establish a working group to coordinate efforts across the National Institutes of Health to cure liver cancer and liver disease, including research into such cases resulting from hepatitis B. The institute may also award grants for developing preventative and diagnostic measures and supporting experimental treatment options. The bill further establishes grant programs, administered by the Centers for Disease Control and Prevention, to support liver disease prevention services, diagnostics, and hepatitis tracking programs and to raise awareness about liver disease within high-risk communities. The renamed National Institute of Diabetes and Digestive, Kidney, and Liver Diseases also must prioritize programs addressing liver disease, including (1) establishing a national clearinghouse for patient data, and (2) developing or expanding the centers for research of liver disease.