This is False!
No credible source uses tangible data, studies, and scientific reasoning to come to this conclusion. The source cited in the original post doesn't even assist this point; it goes over difficulties someone who has a history of breast cancer may face when taking the vaccine. This may include multiple doses, more accommodations with your doctor and others, depending on your history. However, none of those include a chance of getting breast cancer. In fact, the website cited actually mentions that "There’s no link between the COVID-19 vaccines and breast cancer risk or developing breast cancer".
The National Library of Medicine wrote a paper on the effects of the COVID-19 vaccine on people who already have breast cancer. It concludes that it may have some effects, but the patient should consult their doctor. However, it does not indicate in any way that the vaccine causes breast cancer.
There was an article expressing that the COVID-19 vaccine and its variants cause different types of cancer. However, its evidence is not credible, and it's not from a credible source.
https://www.breastcancer.org/managing-life/staying-well-during-covid-19/vaccine-facts
https://pmc.ncbi.nlm.nih.gov/articles/PMC8547943/
https://biomarkerres.biomedcentral.com/articles/10.1186/s40364-025-00831-w