Can a mRNA vaccine alter someone's DNA?...

No. Dr. Paul Offit at the Vaccine Education Center explains how mRNA vaccines are not able to alter a person’s DNA for 3 reasons:

1. Location – mRNA is active in the cytoplasm of a cell, whereas DNA is protected in the cell’s nucleus. The mRNA can NOT enter the nucleus, so the two nucleic acids are never in the same place in the cell.
2. Process – mRNA is not DNA. So, if a person’s DNA was going to be altered, the RNA would have to be made into DNA. This would require a special enzyme which only exists in some viruses. Coronaviruses are not one of them as they have only single-stranded RNA which means that when they enter into a cell’s cytoplasm they don’t need to be translated. Proteins (like the spike protein) can be made directly from the RNA.
3. Stability – mRNA is not very stable and can only stay alive in human cells for hours.

Watch this video where Dr. Paul Offit explains how COVID-19 vaccines based on mRNA technology work.