Rural Health Innovation Act of 2021 This bill establishes two grant programs to increase access to emergency care in rural areas. The Federal Office of Rural Health Policy within the Health Resources and Services Administration must manage these programs. First, the office must provide grants to federally qualified health centers and rural health clinics to enhance the provision of walk-in urgent care, triage, and other emergency services in rural areas. Second, the office must provide grants to health departments in rural areas for emergency services (including triage and transport to emergency departments), primary care services, and similar services provided by emergency departments. Recipients of these grants may use funds for staffing, equipment, and other necessary expenses to provide emergency services.