Vaccine Injury Compensation Modernization Act of 2021 This bill increases the compensation available under the Vaccine Injury Compensation Program for vaccine-related deaths and injuries and otherwise modifies the program. This program provides compensation through a no-fault alternative to litigation for injuries and deaths caused by certain vaccines. Specifically, the bill increases the compensation for a vaccine-related death and the maximum compensation for pain, suffering, and emotional distress from a vaccine-related injury from $250,000 to $600,000. It annually adjusts these amounts based on a consumer price index. In addition, the bill extends from 36 months to five years the period of time after the onset of symptoms of a vaccine injury during which an individual may file a claim for compensation. Furthermore, the Centers for Disease Control and Prevention (CDC) must update the Vaccine Injury Table, which lists and explains injuries and conditions that are presumed to be caused by vaccines, within six months of recommending a vaccine for routine administration to children, adults, or pregnant women. Current law requires the CDC to update the table within two years of recommending a vaccine for routine administration to children.