I’ve always wondered how many people every year develop cancer because their immune systems have been diverted by being challenged by some vaccination that may only be a crapshoot in terms of its efficiency. I prefer to take my chances with the flu, although I’ve never had it.
Zero. Cancer is not caused by the immune system functioning the way it evolved to function.
Whether or not you have received a vaccine, your immune system is busy defending your body against potential pathogens every single day. It is constantly making B and T cells to fight off pathogens, because unless you are a "bubble boy" who must live in a sterile environment due to complete absence of immune system, you encounter pathogens every day.
Cancer can be caused by exposure to some chemicals or pathogens that enter the cell. In either case, they disrupt cell function and/or damage the DNA (which permanently disrupts cell function). Cancer can also be caused by spontaneous DNA mutations (i.e. damage) in cells that the normal DNA repair mechanisms fail to correct.