A vital organ was any organ which was absolutely essential for an individual to live, removal or significant damage resulting in death.
In 2152, the individuals stored on the automated repair station had working vital organs, but had suffered severe neurological damage. (ENT: "Dead Stop")
After the body of James T. Kirk (inhabited by Sargon) had been killed in 2268, Doctor McCoy managed to revive the body and get the vital organs working again, while the consciousness of Kirk was still located outside his body. (TOS: "Return to Tomorrow")
The same year, when the ship was infested by the Beta XII-A entity, McCoy reported that even sword wounds in vital organs were healing at an astonishing rate. (TOS: "Day of the Dove")
When Jono stabbed himself in 2367, no vital organs or major arteries were hit, allowing him to make a full recovery. (TNG: "Suddenly Human")
In 2371, Chakotay comapared the location of the omicron particles within a nucleogenic cloud being to a vital organ. (VOY: "The Cloud")
After Jadzia Dax hat been hit by a Jem'Hadar rifle in 2374, no vital organs were affected, but the rifle had left an anticoagulant in her system. (DS9: "Change of Heart")