Here's a primer for doctors put out by the Children's Hospital of Philadelphia, with instructions on how to "educate" parents over concerns these vaccines could modify DNA-

Can mRNA vaccines alter a person’s DNA?

The short answer to this question is “no,” but in and of itself that is not very satisfying, so let’s take a closer look. We can be confident that mRNA vaccines are not able to alter a person’s DNA for three reasons:

Location — mRNA is active in the cytoplasm of a cell, whereas DNA is protected in the cell’s nucleus. The mRNA cannot enter the nucleus, so the two nucleic acids are never in the same place in the cell.

Process — mRNA is not DNA. So, if a person’s DNA was going to be altered, the RNA would have to be made into DNA. This would require an enzyme called reverse transcriptase. People do not have this enzyme, and only some viruses do. Coronaviruses are not one of them. Viruses in the Coronaviridae family have single-stranded RNA, known as “positive-sense” RNA. This means when coronaviruses enter the cytoplasm, they do not need to be transcribed. Proteins can be made directly from the virus’s positive-sense RNA. In this manner, the mRNA vaccine acts the same way, delivering mRNA to the cytoplasm for direct translation into protein.

Some have suggested that people infected with human immunodeficiency virus (HIV) have reverse transcriptase, and wonder if, therefore, the vaccine could cause changes in their DNA. This is not likely either since HIV infects CD4+ T cells, and the mRNA vaccines are processed by dendritic cells. Likewise, if the presence of reverse transcriptase from a long-term infection allowed for RNA from viruses to alter an infected individual’s DNA, this would have been identified, given the number of people living with HIV.

Stability — mRNA is not very stable. Its half-life in human cells is estimated in hours. For purposes of mRNA-based therapies, modifications have been developed to keep the molecule in cells long enough to allow for the therapy to be successful. But even with this, the mRNA will not lead to protein production for more than 10-14 days.

https://www.chop.edu/news/news-views-3-questions-you-will-get-about-new-mrna-vaccines

So as far as the process goes, it's been determined SARS-CoV-2 DOES have the reverse transcriptase enzyme these doctors swore it didn't have, based on the two studies I referenced above, which is the very reason those studies were so controversial.

There IS a possibility these vaccines can alter DNA based on the same mechanism discovered in SARS-CoV-2. There is no proof of that hypothesis because it hasn't been studied and will most assuredly not be funded by any of the pharmaceutical companies who have no problem injecting every man, woman, and child on the planet with these vaccines.

Also, what happens if somebody who's vaccinated gets infected with SARS-CoV-2 or another virus that has the reverse transcriptase enzyme? It's only logical DNA could be modified in those instances.