Formaldehyde is used in the preparation of many vaccines (not just the flu vaccine) currently available for adults and children. Formaldehyde kills the virus in order to prevent live virus from being injected (and therefore causing flu syndrome). This is an important step in vaccine processing. I am not aware of the amount of formaldehyde that remains in the flu vaccine.
We are exposed to formaldehyde in several different ways every day of our lives. Formaldehyde resins are used in many carpets, furniture, some wood floors, textiles (permanent press), paints and adhesives. Formaldehyde is emitted from burning cigarettes and other burned materials. It is also found in some cosmetics. Too much exposure is certainly a health hazard.
As far as I know, the formaldehyde content of the flu vaccine has not been implicated or reported as a significant health risk. This site contains more information about formaldehyde.