This is an open-ended question which invites speculative scenarios - most and hopefully all of which will have been carefully considered during vaccine development. In concept the vaccine could alter DNA in several very rare and unusual ways, such as if the recipient has HIV and the vaccine RNA became a substrate for the viral reverse transcriptase. More plausibly, the recipient might produce an RNA transcript (possibly produced by a transposable element of viral origin) which base-pairs with the vaccine RNA. This would set off cellular antiviral alarms that affect histone proteins, which tightly adhere to the DNA and can affect the activity of specific genes for a long period of time ( https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5651755/ )
While that sounds very serious, we have to bear in mind not only the balance of probabilities, but also that DNA can be modified in ways that may be permanent for an individual and even transmitted to children and grandchildren - by a vast and mostly unknown set of common environmental circumstances such as famine, obesity, pollutants, and smoking. Look up search results for https://pubmed.ncbi.nlm.nih.gov/?term=transgenerational+methylation+human and make up your own mind.
Anything is possible in biology. If you can't grow a warp drive in your belly, it's only because of physics. Yet many things that could happen are so unlikely that they never do, and we often need to make pragmatic choices between the known risk of death and debilitation on one side, and completely unproven scenarios on the other.
Note: per the question, we're not discussing the original conspiracy video's CRISPR idea, which describes a gene therapy product rather than a vaccine. You could put rat poison in a vaccine, but we wouldn't talk of the safety of rat poison when considering the safety of "vaccines".