This is not even remotely true. If you'd like to read an actual study that shows the opposite, here you go from the CDC: https://www.cdc.gov/mmwr/volumes/71/wr/mm7104e1.htm?s_cid=mm...
Studying infections as they occur is extremely difficult. People who get vaccinated take greater precautions than those who don't. People who get infected take greater risks, and those who are unvaccinated and have been infected are the greatest risk takers of all. Breakthrough infection rate has more to do with cohort behavior than efficacy of natural vs vaccine.
The biggest advantage a vaccine has is repeated exposure to viral material without illness and risk of death. But ultimately the vaccine is just trying to trick your body into creating natural immunity.