This is a false statement.
Here's an article debunking these kinds of claims: https://www.reuters.com/article/uk-factcheck-covid-19-vaccin...
From the article: "Mark Lynas, a visiting fellow at Cornell University’s Alliance for Science group, debunked the idea that a DNA vaccine could genetically modify an organism. Lynas told Reuters that no vaccine can genetically modify human DNA.
“That’s just a myth, one often spread intentionally by anti-vaccination activists to deliberately generate confusion and mistrust,” he said. “Genetic modification would involve the deliberate insertion of foreign DNA into the nucleus of a human cell, and vaccines simply don’t do that. Vaccines work by training the immune system to recognize a pathogen when it attempts to infect the body - this is mostly done by the injection of viral antigens or weakened live viruses that stimulate an immune response through the production of antibodies.”"