Tribal Elder Care Improvement Act of 2019 This bill directs the Department of Health and Human Services to award grants to tribal organizations for providing supportive services (e.g., transportation, case management, and health and wellness programs) to older American Indians, Alaska Natives, and Native Hawaiians. The bill also reauthorizes through FY2024 grants for the delivery of supportive services, including nutrition services, to older American Indians, Alaska Natives, and Native Hawaiians. Additionally, the bill specifies that the technical assistance and training programs provided to such grantees may include program management, data development and use, grant development, and staff training and certification.