The question does indeed rely on the possibility of a general cancer preventive vaccine, something that at the moment seems very unlikely to be possible (there is some research on this and on preventive vaccines for specific types of vaccines). As the link in post #1 implies, it is not impossible, but much more work is needed.
However, the question about using a cancer vaccine, puts the vaccine debate in a different context: based on a disease which we all recognize as a serious danger in our children lives. The debate about current vaccines deals with many diseases that the vaccines themselves helped dramatically reduce or eliminate. These diseases caused many generations ago the fear that cancer causes on us these days. Now that these diseases have been greatly reduced, society has focused on the side effects of these vaccines, and rightfully so. The problem is when we refuse to acknowledge what the experts say about it, and instead focus on subjective opinion.
Anyway, a cancer vaccine would be something I would consider to give my kids. Of course I would look at the research evidence, and based on it, I would consider if the benefits outweigh the risks. I'm not sure if that's the case with the Gardasil vaccine for example. But it would be quite something if a more beneficial than harmful general cancer vaccine would ever come to be. According to the numbers in post 11, at least when it comes to a general cancer preventive vaccine, the chances of harm would need to be very high to not make the vaccine worthwhile
For some specific cancers, well breast cancer affects 1 in 9 women in the western world, so a breast cancer vaccine would have to be quite risky to decide against it too