Vaccines actually do edit the genetic code, but only of some select B-cells. And so does any other antigen that your body develops antibodies for. The process is very complicated and beautiful, and is pretty much exactly the same as in plenty of other multicellular life forms.
If you've ever wondered what's the "storage format" for all the recipes for all the different antibodies that your body produces, that is how does your body even remember infections from years ago - it's DNA, and the recipes are generated via a massively sped up internal evolution, mediated by T cells. The process of maturing B-cells is kind of similar to machine learning. The matured B-cells are then safely stashed away in the bone marrow. Really remarkable stuff.
(I understand that Linus meant that the vaccine mRNA is not reverse transcribed into the host DNA, and that is of course the case. Also, a side note: the human genome is littered with viral genetic material accumulated over time, god knows if it's for good or for bad, we're still too dumb to answer such questions)