Biological Implant Tracking and Veteran Safety Act of 2019 This bill requires the Department of Veterans Affairs (VA) to adopt and implement a device identification system for use in identifying biological implants intended for use in medical procedures conducted in VA facilities. The VA may adopt the system developed by the Food and Drug Administration (FDA) or may develop its own comparable system. Additionally, the VA shall implement a compatible system for tracking the biological implants from human donor or animal source to implantation. The bill requires the VA to implement compatible inventory controls so patients can be notified if they have received a biological implant in a VA facility that is subject to a recall. The bill prescribes requirements for vendors from which the VA may procure biological implants of human and nonhuman origin. Under the bill, the VA shall (1) procure such implants under General Services Administration Federal Supply Schedules unless they are not available under such schedules, (2) accommodate reasonable vendor requests to undertake specified outreach efforts to educate VA medical professionals about the use and efficacy of implants, and (3) procure biological implants that are unavailable under such schedules using competitive procedures in accordance with the Federal Acquisition Regulation. In procuring biological implants, the VA must permit a vendor to use any of the accredited entities identified by the FDA as an issuing agency. Certain biological implants may be temporarily procured by the VA without relabeling under the standard identification system.