There's no public evidence the vaccine caused deaths as claimed in an article published on social media.
The claim cites the government's rough list of unverified reports: Vaccine Adverse Event Reporting System. Anyone can report events to VAERS, and the reports—on their own—cannot be used to say whether a vaccine caused a particular adverse event.
The Centers for Disease Control and Prevention follows up on any VAERS report of death to determine whether it was caused by the COVID-19 vaccine or was coincidental.
According to the CDC, "A review of available clinical information including death certificates, autopsy, and medical records revealed no evidence that vaccination contributed to patient deaths."