Yeah, it's not irony. Fined for inappropriate use of irony.
Flu shots sometimes make you sick initially after you take it, because they give you weakened or dead cells of the disease it's protecting you against in order for you to build antibodies against it so that your body can kill it easily in the event you get a full blown disease.
As such, vaccines are highly specific to that disease you are taking them for, such as the flu. They do not prevent you from catching other things that could make you sick, such as the common cold. You can still get sick if you get a vaccine, vaccines just protect you from dangerous diseases, like the Flu, which can kill you, and there are other vaccines for even worse diseases that are out there.
Also, as nabber pointed out sometimes people are allergic, in which case the need to get a shot that isnt made with whatever it is they are allergic to. Most flu shots are made with eggs, so if you are allergic to eggs, you cant get them. You have to use a vaccine that is made differently, or risk going without.
It is not at all ironic that you got sick the "single time" you got a flu shot. Vaccines are to prevent you from dieing in the event you happen to catch the disease, they do not prevent you from being exposed to people who are sick or carriers of the disease. Whether you are exposed to the disease or not in order to catch it is all luck and where you live. Vaccines help you fight the disease once you fall sick, they dont prevent it.