The correct answer is D. Adaptive immune system.
Vaccines stimulate the adaptive immune system, in particular the antibody-mediated adaptive immune response.
The adaptive immune system produces antibodies and antigen-specific lymphocytes (B cells, T cells) in response to antigens. Vaccines present antigens to the adaptive immune system, causing it to develop antibodies against specific diseases.
The innate immune system and innate immune responses are non-specific. Antigen-specific antibodies and memory cells are characteristic of the adaptive immune system. The variolated and anrigenic immune systems are not real immune system components.
So vaccines work by triggering the adaptive immune system and antibody production.