The short answer is no.
The answer to why depends on which vaccine you are talking about.
I will start with the mRNA vaccines (Moderna/NIH, BioNTech/Pfizer). The whole concept of mRNA vaccines is that they will be translated in the cytoplasm of cells by ribosomes and never enter the nucleus, so are always in a different subcellular compartment than our genetic material (DNA in the nucleus) so cannot modify it. mRNA is also unstable, it gets degraded in cells after it gets translated into protein. That protein (the S-protein of SARS-CoV-2) is then detected by your immune system to train the immune system to react to a SARS-CoV-2 infection in future.
Then there are the viral-vectored vaccines (J&J and Astra-Zeneca). These use a disabled adenovirus (E1 and E2 deleted) which has DNA that encodes the spike protein inserted into the adenovirus genome. This disabled adenovirus genome then gets into the nucleus of a cell. (The reason for using adenovirus is that it is very efficient for getting DNA into cells, much more efficient than the mRNA vaccines.) So there is now extra DNA in the nucleus in the same place as our genetic material. However (as we talked about in class), adenoviruses do not integrate into the host genome, so do not change our genetic material. There are 2 reasons why adenovirus DNA does not integrate. First the genome has proteins bound to the ends, so it does not integrare. There are also no areas of sequence similarity between the adenovirus genome and our genome so homologous recombination (which is one way that external DNA can integrate into genomic DNA). Finally, this disabled adenovirus does not have the machinery to make more of the virus, so it is also only around for a short time, not as short as mRNA, but still a short period of time.
I hope that this helps.
This is from basic molecular biology that has been studied for over 50 years.
-Ken Stedman