A flu vaccine cannot "cause the flu". It can mimic an immune response similar to the flu, but this response is much weaker than what occurs during an actual infection, where a live pathogen is actively spreading and causing harm. The flu vaccine contains an inactivated ('dead') virus or a component of the virus, meaning it cannot replicate or cause illness in the way a live virus does.
The assumption I started my last comment with is extremely important. I'm assuming we are referring to "the flu" as the disease not the influenza virus itself.
Disease is just a named collection of symptoms, that it. A vaccine absolutely can cause those symptoms, and when they occur together it would meet the definition of the disease. That obviously doesn't mean the vaccine caused an influence infection.
Few disease definitions actually take into account severity of symptoms. There are some examples where we have two named diseases where one is distinguished only by being more severe, but unless I'm missing something the flu doesn't fit into that category.
