Theoretically... The injectable flu vaccine is made from killed flu virus--so no, the "flu shot" can not spread the flu. The nasal spray flu vaccine (flumist) is a live, weakened mixture of 3 different flu viruses. It has been "engineered not to cause the flu", yet one of the precautions is that persons with a weakened immune system should not obtain it. So, theoretically, a person with a weak immune system might.
Flu vaccine . All flu vaccine that are administered as shot are considered killed flu vaccines and you can not spread the flu infection from getting the vaccine. There is a live flu vaccine that is a spray in the nose and that theoretically can be spread.