If by "the flu" you're referring to the disease rather than a specific pathogen, a vaccine can cause the flu. Its common for vaccines to cause symptoms similar to the pathogen, though those symptoms are usually more mild.

Think about it for a second. A vaccine is meant to induce an immune response to allow the body to learn the pathogen before a natural infection. Symptoms are little more than the physical effects of your immune system doing its job and responding to a pathogen. Why wouldn't a good vaccine induce similar, though likely less severe, symptoms?