Liam Goldstein pointed me to the Vaccine Adverse Event Reporting System (VAERS) in the United States.
Healthcare providers [in the United States] are required to report to VAERS the following adverse events after COVID-19 vaccination [under Emergency Use Authorization (EUA)], and other adverse events if later revised by CDC:
The VAERS makes their data available for download at https://vaers.hhs.gov/data.html.
Over 109 million doses of COVID-19 vaccines were administered in the United States from December 14, 2020, through March 15, 2021. During this time, VAERS received 1,913 reports of death (0.0018%) among people who received a COVID-19 vaccine. CDC and FDA physicians review each case report of death as soon as notified and CDC requests medical records to further assess reports. A review of available clinical information including death certificates, autopsy, and medical records revealed no evidence that vaccination contributed to patient deaths. CDC and FDA will continue to investigate reports of adverse events, including deaths, reported to VAERS.
Note, from https://www.dw.com/en/fact-check-no-links-found-between-vaccination-and-deaths/a-56458746
As far back as 2015, a study assessing claims of deaths from vaccinations highlights that data from the VAERS system is skewed, as it is a system which "accepts any submitted report of an adverse event without judging its clinical significance or whether it was caused by a vaccination." The Centers for Disease Control and Prevention also warns against reporting bias and inconsistent data quality in the VAERS system.