The mRNA vaccine would not alter the human DNA as it never enters the nucleus of the cell.
It would seem that those who post such claims misunderstand genetics. The Pfizer COVID-19 vaccine is an mRNA vaccine (or messenger RNA vaccine). According to the PHG Foundation, a health policy think tank, RNA vaccines work by introducing an mRNA sequence (the molecule which tells cells what to build) which is coded for a disease-specific antigen. Once produced within the body, the antigen is recognized by the immune system, preparing it to fight the real thing.
There are many advantages to mRNA vaccines, including the fact that they can be quick and cheap to develop. Real concerns center around the length of protection they offer, not whether they alter your DNA. Indeed, mRNA vaccines do not do anything to the DNA in a human cell. They just trigger an immune response by instructing the body's cells to produce the vaccine antigens.
The COVID-19 pandemic has given rise to a lot of potentially dangerous misinformation. For reliable advice on COVID-19 including symptoms, prevention and available treatment, please refer to the World Health Organisation or your national healthcare authority.