Think of it a bit like a lottery. There is a small chance every time the virus copies itself and spreads that it will make an error - mutation - that happens to be one that means the vaccines won't recognise it, and perhaps also combined with something that makes it spread more. It's random chance if this will happen or not. If the virus is 'lucky', it only takes one winning ticket to achieve that, and that could potentially happen even with one ticket. But the chances are very small. On the other hand, if it has millions of tickets, the chances of it getting a 'good' one are much higher. It still might not - we could have millions of mutations, and by chance, all of them are fine, or don't spread, or whatever. But we'd have to be quite lucky, if the virus had so many chances against us.
So the lower we can keep transmission, not just here but around the world, the longer time we have before a bad mutation starts to spread. And in that time, we can be continuing to vaccinate people to reduce the chances of the virus getting lucky, and also developing new vaccines.
The dangerous situation is when there is a lot of spread but new vaccines aren't ready, or there are still a lot of unvaccinated people, because then the bad mutations are able to spread and not be traced or noticed as easily.
So it's possible to win - but it takes both luck, and people being sensible about not letting transmission get out of control or thinking that case numbers don't matter. That buys us time - the vaccines will eventually reduce transmission a lot, which will help, and there will also be new vaccines. But the dangerous time is while that is happening.