Accelerating Connected Care and Education Support Services on the Internet Act or the ACCESS the Internet Act This bill provides funding to support distance learning and connected care in response to COVID-19. Specifically, the bill provides funds for (1) elementary and secondary school emergency relief grants to purchase internet-connected devices and associated services that support distance learning, (2) a pilot program to award grants for the purchase and distribution of internet-connected devices for libraries in rural and low-income areas, (3) the COVID-19 Telehealth Program of the Federal Communications Commission, and (4) the furnishing of internet-connected devices and associated services to veterans for accessing telehealth services.