The article has been taken out of context. The author says vaccines lead to the proliferation of immune cells that protect you from future infections.
Salzberg's answer to whether the COVID-19 vaccine changes your DNA is: "Not directly, no. But yes, thanks to your own immune system, the overall mixture of DNA in your body is a tiny bit altered after you get any vaccine. Your DNA is also changed every time you recover from an infection, including the common cold. But the only change is in the DNA of a tiny number of immune cells, which hang around as guardians against future infections."
The cells that "show" our body how to fight the virus comprise a marginal portion of our body. Salzberg says every other cell type, such as the skin, heart, lungs, and brain, is completely unaffected by the vaccine. The cells that do get affected are needed for the body to remember how to combat the virus. Salzberg in his article addressed how international organizations were providing a simplistic, albeit correct, explanation of how vaccines don't change human DNA and dove into how a body and its cells react to infections and vaccinations.
As the World Health Organization has stated, “COVID-19 mRNA vaccine technology has been rigorously tested for safety, and clinical trials have shown that mRNA vaccines provide a long-lasting immune response. mRNA vaccines are not live viral vaccines and do not interfere with human DNA."
The reaction that a body has to the vaccine is based on sounds science, and cannot in any way be classified as gene therapy.