This is false: According to the Paul-Ehrlich-Institut, the Federal Institute for Vaccines and Biomedicines, an agency of the German Ministry of Health:

There is no risk of integration of mRNA into the human genome. In the case of humans, the genome is located in the nucleus of the cell in the form of DNA. The integration of RNA into DNA is not possible, among other things, because of the different chemical structures. Furthermore, there is no evidence that the mRNA integrated by the body cells after a vaccination will be converted into DNA.

According to an article on DW, “generally speaking, no vaccine can genetically modify human DNA, according to experts. Mark Lynas, a visiting fellow at Cornell University’s Alliance for Science group, told Reuters: “Genetic modification would involve the deliberate insertion of foreign DNA into the nucleus of a human cell, and vaccines simply don’t do that.””

People are made of stories, not atoms.