Improving Care for Vulnerable Older Citizens and People with Disabilities through Workforce Advancement Act of 2021 This bill establishes demonstration projects to improve care and service delivery through the use of direct care workers (e.g., nursing assistants and home health aides) for older adults and individuals with disabilities who have chronic illness or are at risk of institutional placement. The Health Resources and Services Administration (HRSA) must award grants to partnerships of health care providers, long-term care facilities, and other organizations to carry out the projects. Specifically, HRSA must award grants for incorporating direct care workers into interdisciplinary care coordination teams to support transitions in care and prevent unnecessary hospital readmissions. HRSA must also award grants for (1) using the abilities of direct care workers to manage and improve the health of individuals with multiple chronic conditions and long-term care needs; or (2) training direct care workers to take on greater responsibilities in managing specific diseases, such as Alzheimer's disease and diabetes.