Mutations can occur with any replication. But the more case transmissions, the more viral replication is ongoing (more hosts).
To prevent variants, we need to reduce mutations. To do that we need to reduce the amount of circulating virus. To do that we need to keep cases to a minimum and we need to do it globally.
Will seek your confirmation of this statement - but my understanding also is that transmission between vaccinated people comes with a much weaker 'viral load' of the virus meaning you have much weaker forms transmitting and mutating, versus having unvaccinated people who would carry a much more potent viral load leading to a much stronger mutation.