No. Right now, there are three kinds of COVID-19 vaccines: mRNA vaccines (Pfizer and Moderna), a protein subunit vaccine (Novavax) and a viral vector vaccine (Johnson & Johnson).
It is true that these vaccines send genetic material to our cells – just like blueprints or instructions, this genetic material teaches our cells how to defend against the coronavirus. However, the genetic material never enters the nucleus of our cells, which is where our DNA is stored.
That means the genetic material provided by the vaccines as defense instructions against COVID-19 does not interact with or affect our own DNA in any way. Moreover, the genetic material from the vaccines harmlessly dissipates shortly after it has done its job.
There’s no risk to your DNA from the COVID-19 vaccines.