I understand this might depend on the types of vegetables, but is there an average or studied specifics? Does it die immediately? Is there a way to precisely diagnose death in plants? If so, what are the conditions for it?
The short answer is that as long as the vegetable/fruit is fresh looking - i.e. the cells have not disintegrated - they will be respiring, many cells will be functioning quite normally, and the plant is still technically alive. In cases where the part of the plant we treat as a vegetable is a part intended for reproduction (e.g. a seed, or a tuber like a potato) the plant will keep growing.
The point at which the plant dies is not clearly defined like it is in animals, but generally if you can still eat it, it's still alive.
Death in plants is quite different from that in animals - we refer to it as senescence. The key difference is that it happens to tissues and organs which can die and separate from the organism. Individual leaves can die without the plant's health being affected. Once this has happened to all the parts, the organism is considered dead, but if there is any respiring tissue left, it's still alive.