Depends on if the ingredients, primarily the RNA sequence, are actually what is listed in the vaccine. I'm pointing out that RNA can alter DNA and that what the OP said is factually incorrect.

Article from May 12 2021: Strongest Evidence Yet Shows SARS-CoV-2 May Insert Itself Into The Human Genome

The study cited:

SARS-CoV-2 RNA reverse-transcribed and integrated into the human genome

Prolonged SARS-CoV-2 RNA shedding and recurrence of PCR-positive tests have been widely reported in patients after recovery, yet these patients most commonly are non-infectious1–14. Here we investigated the possibility that SARS-CoV-2 RNAs can be reverse-transcribed and integrated into the human genome and that transcription of the integrated sequences might account for PCR-positive tests. In support of this hypothesis, we found chimeric transcripts consisting of viral fused to cellular sequences in published data sets of SARS-CoV-2 infected cultured cells and primary cells of patients, consistent with the transcription of viral sequences integrated into the genome. To experimentally corroborate the possibility of viral retro-integration, we describe evidence that SARS-CoV-2 RNAs can be reverse transcribed in human cells by reverse transcriptase (RT) from LINE-1 elements or by HIV-1 RT, and that these DNA sequences can be integrated into the cell genome and subsequently be transcribed. Human endogenous LINE-1 expression was induced upon SARS-CoV-2 infection or by cytokine exposure in cultured cells, suggesting a molecular mechanism for SARS-CoV-2 retro-integration in patients. This novel feature of SARS-CoV-2 infection may explain why patients can continue to produce viral RNA after recovery and suggests a new aspect of RNA virus replication.

Supposedly they think this might be incorrect and due to genetic artifacts, but who knows. It's molecular biology and requires further study and validation to determine what is happening. Obviously those people are still shedding the virus despite no longer being infectious, so the question is why.

https://www.purdue.edu/newsroom/releases/2021/Q2/the-covid-19-virus-may-not-insert-genetic-material-into-human-dna,-research-shows.html