The virus mutates about every 2 infection cycles. This is know from many many genetic sequences that have been taken of the virus from many patients. That may sounds like a lot, but that is one change in a genome of 30,000 bases Importantly, those changes have not occurred in the particular parts that would effect the vaccine. There are a few proteins (spike and nucleocapsid) that are recognized by the immune system. Additionally, there are only parts of those proteins (epitopes) that the immune system cares about. This is a tiny portion of the genome.
There are a lot of changes that have no effect on the virus. For those that don't remember their biology/genetics, proteins are made up of 20 amino acids, and the sequence of amino acids is coded by our DNA which is made up of 4 bases. The amino acid sequence is dictated by codons of 3 bases. There is a lot of built in redundancy as 3 bases is 64 different combinations to encode 20 amino acids (and stop codons). So some mutations don't even change the amino acid. Other mutations swap one amino acid for another very similar amino acid that has minimal effect on the protein, and still other mutations occur in non-coding regions (not in one of the codons).
There are a lot of mutations (perhaps most mutations) that will have no effect on the virus. Since there are several other proteins, there are mutations that will alter the virus in other ways but not the immune response. And even mutations that change the amino acid sequence of S or N may not effect immunity if they don't change one the epitopes.
And finally, your immune system makes lots of different antibodies to a virus recognizing a variety of epitopes. A change to one epitopes may only effect one type of antibody and your immune system can still recognize the virus is a lot of other ways.
And one final comment, polyclonal antibodies to SARS-CoV spike protein recognize the spike protein of SARS-CoV2 which causes COVID-19. That is at least 17 years worth of divergent mutation in the virus in the bats that doesn't alter immunity. Probably more because SARS supposedly passed from bats to civets to humans.
