I work in neuroscience, so I cannot speak as much to what cancer researchers do. I have seen little lab-based primate research save for some small specialty fields in neuroscience like grabbing stuff and chit-chatting fine motor coordination and language acquisition. Even then, the research is on very small primates like macaques. Thinking practically, feeding thousands of giant apes every day just waiting for one to get breast cancer sounds very expensive and very boring.

The typical procedure for cancer research is to either mutate a mouse and breed them or to inject a human tumor seed into the mouse, causing the animal to get cancer. These processes are have been very well-described in other comments. I would like to add that in neuroscience we use a more diverse range of animals in the lab, but also many researchers go out in the field and observe animals in the wild. Studies on things like tool usage and language acquisition are in fact conducted by watching animals (usually dozens not thousands) and seeing what they do. We save money by having them–and sadly, sometimes the researchers–forage for their own food.

I will say that you have pointed out a systematic flaw with biomedical research, which is that researchers have a much easier time studying a disease caused by particular mutation(s) than one with a more complex etiology. It is much simpler (and cheaper) to make a mouse model that has a BRCA1 cancer or some other genetic disease (e.g. Huntington's) than a disease that is complicated and hard-to-measure (e.g. Major Depressive Disorder), so research funding tends to gravitate towards these projects, even if the diseases they study impact many fewer people.