I think the phrase vie sociale would be understood, but it is not idiomatic. I would say
Il n’a pas de vie.
The meaning is subtly dependent on the context. For example, “ce n’est pas une vie” means “this is no way to live”, i.e. there’s something wrong with my lifestyle (which is usually not something I can fix); while the absence of a social life is often an aspect of this, it is not the primary meaning.
I think I would more spontaneously use other ways of phrasing “he doesn’t have a social life”, not using the word vie:
Il ne sort pas.
Il reste dans son coin.