End Diaper Need Act of 2019 This bill establishes various programs to increase access to diapers for low-income families with infants and toddlers. Specifically, the bill provides grants to agencies or organizations with experience in community distribution services for the purchase, warehousing, and distribution of diapers and diapering supplies (e.g., diaper wipes and creams) to families with (1) at least one child under the age of 4, and (2) an income of no more than 200% of the federal poverty level. The bill also permits states to use Medicaid funds to provide diapers and diapering supplies to low-income families with a child 3 or older who has been diagnosed with one or more specified forms of conditions that create a need for the use of diapers. Additionally, the bill includes diapers and diapering supplies as qualifying medical expenses under (1) health and medical savings accounts, and (2) flexible health spending and reimbursement arrangements.