Leave no Veteran Behind Act This bill requires the Department of Veterans Affairs (VA) to seek to contact covered veterans to encourage them to receive comprehensive physical, mental health, eye, and audiological examinations. A covered veteran is a veteran who is enrolled in the VA health care system and has not received health care furnished or paid for by the VA in the past two years. Under the bill, if a covered veteran elects to receive more than one of the comprehensive examinations at a VA health care facility, the VA shall seek to furnish all such examinations on the same day. A covered veteran may also receive an examination from another specified health care provider. The bill provides that the VA may pay for a rural covered veteran to travel to a health care facility to receive a comprehensive examination. The VA shall seek to enter into agreements with non-profit organizations to provide shuttle service to rural covered veterans for such examinations.