The CDC says, “COVID-19 vaccines do not change or interact with your DNA in any way.”
https://www.cdc.gov/coronavirus/2019-ncov/vaccines/facts.html
The writer says, “Not so fast,” and gives what appears to (uneducated) me to be a credible and informative explanation.
Just to be clear, I see the article as interesting in a purely academic sense. I don’t recommend seeking advice in it.