My understanding is that cancer is one of those things you die from if you don't die at child birth, get eaten by predators, or get the flu, polio, malaria, the plague, etc.
That is, the reason cancer is such a big deal in modern times is because other causes of death have been controlled.
I don't know much about elephants but I imagine cancer is still low on their list of things to worry about (humans are probably a lot higher up) If they started living a lot longer, it would probably show up more.
People under 40-50 also rarely get cancer. Although I'm sure the human environment has produced some more causes of it that make those rates higher than they used to be.