The answer to your question (from the article):
"The vaccine mRNA itself is unlikely to somehow integrate into human genes, as that process would require the mRNA to use an enzyme called “reverse transcriptase” to turn itself into DNA before being integrated into human genes. Instead, the more likely avenue comes because Pfizer and Moderna use what scientists call “DNA plasmids” - a circular DNA molecule - to make bacteria produce the mRNA that is the active ingredient in the vaccine."
"The manufacturing process inevitably leads to a small amount of DNA plasmid contamination in the mRNA strands the bacteria make. No one had ever made mRNA at the speed and scale used in the Covid vaccines, requiring the manufacturers essentially to invent processes in a matter of months in 2020."