It is currently unknown whether this occurs, though it is considered unlikely and unimportant. There is a plausible mechanism which could cause it, but as of February 2022, it seems no experiments have been done to prove or disprove whether this actually occurs.
The mechanism is LINE1-mediated retrotransposition, in which a retrotransposon (which make up 17% of our DNA) is translated into a reverse transcriptase protein and picks up the vaccine RNA to reverse-transcribe back into DNA and incorporate into the genome of the cell. (Normally these reverse-transcribe their own RNA, but they can occasionally integrate other RNA instead. This is known to have happened long ago with (non-retroviral) RNA viruses, for instance.)
Studies have tested whether this occurs with the coronavirus itself, and although one found that it did, it is not peer-reviewed in the usual way and is considered to be a false positive, an artifact of the test method. Follow-ups have found no evidence that coronavirus integrates into the genome. (However, the vaccine is not the virus, so this is a clue, not proof.)
Even if this does happen, it most likely will just result in the cell producing spike protein continuously, which will trigger the immune system to kill the affected cell.
While it is possible for retrotransposons to cause cancer, they already do this on their own, without needing a virus or vaccine at all, so you are already at risk of it just by being a mammal.
Flu RNA Vaccine: A Game Changer? (December 2020)
Then, even if the risk of genomic integration is widely considered as null and is not a biosafety concern, eukaryotic cells have been shown to be able to provide, to some extent, reverse transcription activity [25 –28]. Further research on that eukaryotic reverse transcription activity, in the context of RNA vaccination, might be of interest for the scientific community .
Do RNA vaccines obviate the need for genotoxicity studies? (November 2020)
The potential for exogenous RNA, viral or other, to integrate into human DNA from vaccination is at best theoretical at this time. The potential for any such RNA to fundamentally drive altered or oncogenic processes in human cells is also theoretical. Indeed, some may argue that the infections from RNA (DNA) viruses better present viral RNA (or DNA) into the cellular locations where integration or first-hit events occur more than any potential vaccine. Until any such hypothetical concerns are tested or observed from vaccination or natural infection, regulatory safety assessments of RNA vaccines should include genotoxicity studies.
mRNA vaccines: Why is the biology of retroposition ignored? (July 2021, preprint)
I conclude that is unfounded to a priori assume that mRNA-based therapeutics do not impact genomes, and that the route to genome integration of vaccine mRNAs via endogenous L1 retroelements is easily conceivable. This implies that we urgently need experimental studies that would rigorously test for the potential retroposition of vaccine mRNAs. At present, the insertional mutagenesis safety of mRNA-based vaccines should be considered unresolved.
This image of the process is from the above preprint: