Goldilocks,
I see where you’re going, but it’s not likely. The only reason there is a pandemic is because there is a significant group of people who are at high risk if they get infected. If the vaccine prevents those people from getting sick, then it doesn’t really matter that the virus might be spreading. It would no longer be a crisis. At worst it would just be like the common cold.
Also, even if vaccinated people can spread the virus, they probably spread it less than sick people since they get rid of it more quickly.