I believe for Pfizer the preclinical DART and teratogenicity studies in rats are still ongoing, but nobody expects the readouts to be anything bad. That was in the last update I saw, so my info may be out of date.
Don't almost all chemical carcinogens basically raise the probability you will get cancer while the carcinogen is present in you, with higher concentrations raising the probability more? And so your cancer chances are roughly the integral over time of the concentration of the carcinogen within you?
For either of these vaccines even if some part of the delivery/packaging mechanism, such as the lipid enclosure, is carcinogenic, you are only getting a small amount and only getting two short exposures. It seems like that would make the cancer rate from them so low as to be almost impossible to detect.
As you said, the exposure amount is quite low, so barring something really toxic, I wouldn't expect a danger from chemical reactions.
However, the body is a complex system; if something in the vaccine triggers a change in function, that could lead to side effects that are only detected years later; most likely through epidemiologic studies comparing those who got the different types of vaccines with those who didn't.
Of course, the disease itself has potential long term effects as well; and has already been studied to have significant medium term effects.
Maybe in addition to offering immunity to covid-19 it also changes your immune response to a type of cancer cell. Then it's found that people with this change almost always suffer from the same type of cancer.