I say it's misinformation disguised as caution. I found it hard to find anything that credibly contradicts the following: Will a COVID-19 vaccine alter my DNA?
No. COVID-19 vaccines do not change or interact with your DNA in any way.
Both mRNA and viral vector COVID-19 vaccines deliver instructions (genetic material) to our cells to start building protection against the virus that causes COVID-19. However, the material never enters the nucleus of the cell, which is where our DNA is kept.
Also I just finished studying for a written drivers license exam that they told me, after checking my driving record at the exam center, that I didn't have to take because no moving violations in decades. I was hoping the guy sitting next to for the driving exam would ask me some questions.
So I recognize the following to mean 'Yield, Do Not Enter Double Helix Zone.