I know several breeders that can tell you every major illness suffered by every puppy that has left their property. So - while they may not have the health records of the siblings of their dogs, they can tell you if cancer is an issue in their lines.
Do they voluntarly stop breeding and sterilize all dogs in those lines they kept for breeding where there is an elevated incidence of cancer? Without the history of the siblings, parent, and grandparents of the mating pair, they are also in the dark genetically.

Shall we do a very quick search on Golden Retriever cancer breeding? An interesting link first on the list is: Golden Retriever Club of America. What is the closest they come to addressing the issue of cancer? In one section they discuss on the breeding decision for those who own a Golden:

SOUNDNESS. Your dog should be tested free of certain genetic defects, as should the proposed mate. Knowledge of the status of parents, grandparents, siblings, etc. with regard to genetic testing is also desirable. HIPS should be properly X-rayed, and the X-rays submitted to the Orthopedic Foundation for Animals or PennHIP to be read as free of hip dysplasia. HEARTS should be examined by a board-certified cardiologist. EYES should be examined annually by a board-certified veterinary ophthalmologist and be free of hereditary cataracts, progressive retinal atrophy, and any other eye anomaly. ELBOWS should be properly X-rayed and the X-rays submitted to the Orthopedic Foundation for Animals or a board-certified veterinary radiologist, to be read as free of elbow dysplasia.

Any inheritable defects, including but not limited to retained testicles, overshot or undershot jaw, congenital heart defects, recurrent skin problems, thyroid deficiency, immunological problems, orthopedic problems and recurrent seizures or epilepsy occurring in either parent are all reasons not to breed, regardless of other qualities.

Criteria for breeding a Golden?

Note the absence of the word cancer. At any point do they make the statement: If any parents, grandparents, or siblings have cancer any offspring will not qualify for their papers?

They do recognize the problem of cancer is epidemic in the breed.

As you know, Golden Retrievers have a high incidence of cancer, with a 1998 Health Survey indicating that 57% of females and 66% of males will be affected. While many types of cancer are elevated in the breed, approximately half of the totals are lymphoma (1 in 8 Goldens) and hemangiosarcoma (1 in 5 Goldens).

What is the near term purpose of their research efforts:

This information has the potential to impact therapy decisions, as owners and veterinarians can take breed specific data into consideration when they explore treatment options...In addition to financial support, we also aid researchers in obtaining vital tissue and/or blood samples from affected dogs.

Golden Retriever Club response

It sounds noble that they are supporting research for treatment, but that's downstream and after the fact.

In summary you have a major organization who although they have made progress in combating hip dysplasia, have made no effort to provide strict breeding guidelines to prevent the propagation of oncogenes from one generation to the next. There's no other way to say it: Follow the money trail. If the incidence is 57% of females and 66% of males will be affected imagine an organization coming up with a program where their breeder members are told they can not breed 60-70% of their breeding stock starting tomorrow based on family history? It's not going to happen any time soon. Maybe when the breed is up to a 95% incidence of cancer the issue will be forced. In the mean time the oncogenes will continue to propagate not just in the pure breeds be it Goldens, Labs, Boxers etc, but also into the mix breeds.