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When I got the vaccine, I had that same doubt too, but no - mRNA doesn't enter the cell nucleus where DNA is located, so it doesn't mess with your genetics. The vaccine works in the cytoplasm and then gets naturally broken down.
The panic about mRNA and DNA comes from lack of information, really, because the molecule is way too big to cross the nuclear membrane - it stays working in the cytoplasm, produces the spike protein, and then your body naturally breaks it down without leaving any trace in your DNA. My dad spent days reading conspiracy theories on the internet before getting vaccinated, but once I explained this to him he understood that it was physically impossible to alter genetics that way, since mRNA can't even get close to the nucleus where DNA is located.
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