This claim is false.
Cancer Council an Australian non-profit organization focusing on cancer states, "Microwaves, radio waves, and the light we can see, are all examples of non-ionising radiation." Then it claims that the only non-ionsing radiation that causes cancer is ultraviolet (UV) light such as staying in the sun for too long without protection, therefore microwaves are not known to cause cancer.
CancerCouncil
The FDA directly states in a section on their website about microwave use, that "microwave energy is changed to heat as it is absorbed by food, and does not make food “radioactive” or "contaminated." Making it apparent that the direct use of microwaves does not cause cancer.
FDA
Although these sources have led to the conclusion that the direct use of microwaves has no correlation with developing cancer. The website attached to this claim may have misinterpreted claims made by doctors, instead of microwaves themselves posing threats of developing cancer such as colon cancer, what actually is being microwaved such as different materials of tupperware, containers, etc., may be the problem.
Moffitt Cancer Center in Tampa, Florida, a nonprofit cancer treatment and research center makes it apparent that plastic food wraps, containers, etc., labeled by the FDA as 'microwave-safe,' are safe to use and are not known to cause cancer. The article continues on to say that the public is worried about the toxin called, 'dioxin' entering food, which is commonly associated with microwaving plastic products. The cancer center makes it clear that "However, plastics don't contain dioxins. Rather, dioxins are created when plastics and other substances burn or melt. As long as you don't burn your food in a microwave or use containers not intended for microwave use that could melt and possibly leak chemicals into your food, you aren't exposing yourself to dioxins. If a container does not have a microwave-safe label, it isn’t necessarily unsafe; the FDA just hasn’t determined whether it is or not.” The article makes it clear that plastic doesn't naturally contain the toxin called dioxin that can lead to cancer; however, the toxin can be released when plastic burns or melts. This is can be rare for microwave-safe products approved by the FDA.
Moffitt
All the sources I have gathered point to this claim being false as there is no direct association between microwaving and developing cancer, specifically in the colon. Also, toxins do not get released into food by the radiation of microwaves and the plastic containers and wraps used. Therefore this claim is false.