The European Medicines Agency is an agency of the EU that's in charge of evaluating medicines, including vaccines. As part of that responsibility it maintains the EudraVigilance system, which is charged with reporting and evaluating adverse reactions to medicines. I'm not able to tell you how that system works. In the United States reporting on adverse reactions to vaccines is done on a voluntary basis and is coordinated by the FDA and CDC through the VAERS database.
By way of comparison to what follows, the death toll for Covid vaccines in the US as reported in the VAERS database, as of 5/10/21, stands at 4,434.
This evening I found a link at FR which pointed me to this page, which appears to be an anti-vaxxing site:
10,570 DEAD 405,259 Injuries: European Database of Adverse Drug Reactions for COVID-19 “Vaccines”
Those totals are gleaned from the EudraVigilance reports for each of the four vaccines--which you can find by scrolling down to Covid-19 at this page. Note that there are four separate links: