It's been found that through reverse transcriptase it's actually possible for the RNA from covid infections to be integrated into the genes: https://pubmed.ncbi.nlm.nih.gov/33330870/ . If covid itself can do it, can the mRNA vaccines do it too?
More importantly, though, all the paper shows is that under specific lab conditions (which are distinct from conditions in living cells!), viral RNA might integrate into the host genome (subject to the caveats about artefacts). In particular, it requires active retrotransposons [3], which are elements that are inactive in cells in the body. Unless a virus brings its own retrotransposon (and SARS-CoV-2 doesn’t), it can’t integrate into the host genome. But even if SARS-CoV-2 did, they mRNA vaccine definitely doesn’t include a retrotransposon.
[1] https://www.biorxiv.org/content/10.1101/2021.03.05.434119v1 [2] https://www.pnas.org/content/118/21/e2105968118 [3] https://twitter.com/GeneticsMike7/status/1404927629812240388