Toxic Exposure Training Act of 2021 This bill requires the Department of Veterans Affairs (VA) to provide training to its health care and benefits personnel regarding veterans who have been exposed to toxic substances. The VA must provide its health care personnel with education and training to identify, treat, and assess the impact of illnesses related to exposure to toxic substances. Such training must address how to ask for additional information from veterans regarding exposure to different toxicants. Additionally, the VA must establish an annual training program for claims processors who review claims for disability benefits related to service-connected disabilities based on exposure to toxic substances.