Medical and Health Stockpile Accountability Act of 2023 This bill requires the Office of the Assistant Secretary for Preparedness and Response, which is a part of the Department of Health and Human Services, 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, the office 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.