Vaccines reduce overall levels of disease and therefore viral transmission; that's completely false. Transmissibility of SARS‑CoV‑2 can be hard to measure since asymptomatic people can be highly infectious.

COVID-19 vaccination with an mRNA-based vaccine showed a significant association with reduced risk of asymptomatic SARS-CoV-2 infection here: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7989519/