The Pfizer and Moderna vaccines are mRNA vaccines. mRNA does not enter the cell nucleus--it remains in the cytoplasm. There is a paper that studied how mRNA synthesized in the nucleus diffuses into the cytoplasm (after transcription) which may be worth reading. mRNA is used by ribosomes in the cytoplasm to synthesize proteins, enzymes, and other bits--it's how the virus produces more virions with the exception that the mRNA fragments used by both of these vaccines are the sequences that produce the surface receptors.

You cannot modify DNA from mRNA.

What's making people sick in these vaccines is the polyethylene glycol used to stabilize the lipid nanoparticles that encapsulate the mRNA as its delivery system, and PEG are known to cause anaphylaxis. PEG as a component of vaccines is not well-studied and should be viewed with caution, because it certainly does provoke an immune response that can be dangerous if not outright lethal.

If you wanted to go down the rabbit hole of a vaccine that could feasibly modify the patient's DNA, then I'd suggest looking into the AstraZeneca vaccine which uses a modified adenovirus (DNA virus) to inject DNA into the nucleus where the cell transcribes mRNA from the DNA. There's a YouTuber who used a similar adenovirus to "cure" his lactose intolerance for a couple of years by modifying his own DNA in intestinal tissue responsible for producing lactase. Except that the effect wasn't permanent because the modifications could not be transcribed to the daughter cells.

The Pfizer and Moderna vaccines are mRNA vaccines. mRNA does not enter the cell nucleus--it remains in the cytoplasm. There is a paper that studied how mRNA synthesized in the nucleus diffuses into the cytoplasm (after transcription) which may be worth reading. mRNA is used by ribosomes in the cytoplasm to synthesize proteins, enzymes, and other bits--it's how the virus produces more virions with the exception that the mRNA fragments used by both of these vaccines are the sequences that produce the surface receptors.

You cannot modify DNA from mRNA.

What's making people sick in these vaccines is the polyethylene glycol used to stabilize the lipid nanoparticles that encapsulate the mRNA as its delivery system, and PEG are known to cause anaphylaxis. PEG as a component of vaccines is not well-studied and should be viewed with caution, because it certainly does provoke an immune response that can be dangerous if not outright lethal.

If you wanted to go down the rabbit hole of a vaccine that could feasibly modify the patient's DNA, then I'd suggest looking into the AstraZeneca vaccine which uses a modified adenovirus (DNA virus) to inject DNA into the nucleus where the cell transcribes mRNA from the DNA. There's a YouTuber who used a similar adenovirus to "cure" his lactose intolerance for a couple of years by modifying his own DNA in intestinal tissue responsible for producing lactase. Except that the effect wasn't permanent because the modifications could not be transcribe to the daughter cells.