Medical and Health Stockpile Accountability Act of 2022 This bill requires the Department of Health and Human Services (HHS) to develop an automated application to track the amount of supplies in the Strategic National Stockpile and in similar inventories maintained by states, tribes, territories, and private entities (e.g., hospitals). Additionally, HHS must (1) provide for an annual exercise to test the effectiveness of the application, and (2) establish a program to assist health care entities in obtaining automated vendor management systems.