Vegemight was an achievement in Plants vs. Zombies Adventures that was earned by completing a level without having any plants KO'd. Even if they were revived, the player still will not get the achievement.
Its name is a portmanteau of "Vegemite", an Australian paste made from yeast extract and "might", meaning great and impressive power or strength.
You can just start the first level keeping the Peashooters away from the zombies and not letting them get killed as an easy strategy. In the early levels, it was quite easy to earn. However, as progress continues, the zombie toughness and numbers get bigger, making it more difficult.