Contrary to what many have read, an mRNA vaccine won't change your DNA. But it does instruct the recipient's body to produce an antigen that the immune system will recognise. There could be a potential here for autoimmune reactions (see Safety section): https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5906799/
"Potential safety concerns that are likely to be evaluated in future preclinical and clinical studies include...the biodistribution and persistence of expressed immunogen, stimulation of auto-reactive antibodies and potential toxic effects of any non-native nucleotides..."
And in the BMJ: https://www.bmj.com/content/371/bmj.m4347/rr-6
And in another article, under Potential Advantages and Limitations of mRNA-based SARS-CoV-2 Vaccines: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7218962/
Hence why vaccine development normally lasts years, not months.