It seems to me, that given a particular genetic propensity for developing cancer, it is far more likely that one would get it sooner rather than later from eating poorly, or being overweight instead of from trace trace trace amounts of substances found in cosmetics.
I don't think a single case of cancer in humans has ever been reported to be from cosmetic use, and I have scoured the literature for such cases.
I don't think such chemicals are completely innocent, they are tested in massive quantities compared to what people use and they can do harm in such cases. But normal exposures in humans are orders of magnitude lower.
I think the real reason to avoid cosmetics or personal care products is that they are generally unnecessary and their production, packaging, transport, advertising, and manufacture all puts a tremendous and needless burden on the planet. And I strongly suspect that this is the real motivation behind EWG's scare tactics.
In short, I don't agree with the cancer-causing meme, but I do think we should stop trashing the planet, and I tend to use as little as possible. But to spend $9 on the "eco-friendly chemical-free" shampoo that is just as loaded with synthetics but with a more clever marketing ploy than the $0.99 bottle of white rain is missing the point. The point being we should use a lot less. Of everything.