Tribal Medical Supplies Stockpile Access Act of 2022 This bill requires the Department of Health and Human Services (HHS) to deploy drugs, vaccines, biological products, medical devices, and other supplies from the Strategic National Stockpile directly to health programs or facilities operated by the Indian Health Service (IHS), tribes, or tribal organizations. Such supplies from the stockpile are used to respond to public health emergencies. Specifically, if HHS distributes pandemic or epidemic products (e.g., drugs or medical devices) to states or other entities, HHS must also deploy such products directly to health programs or facilities operated by IHS or other tribal entities. HHS must coordinate with states, programs, and facilities to ensure supplies and products are distributed to such tribal entities.