Veteran Overmedication and Suicide Prevention Act of 2021 This bill requires the Department of Veterans Affairs (VA) to contract with the National Academies of Sciences, Engineering, and Medicine to report on the deaths of covered veterans who died by suicide during the last five years, regardless of whether information relating to such deaths has been reported by the Centers for Disease Control and Prevention. A covered veteran is any veteran who received VA hospital care or medical services during the five-year period preceding the veteran's death. Among other elements, the report shall include the total number of covered veterans who died by suicide, violent death, or accidental death, as well as certain demographic information.