In the videogames, its not even a pretend democracy. The overseer is appointed for life by vault tek.

True, I'm not saying they adhere to an actual communinist way of organizing vault society, but the day to day has to be communal and as far as I'm aware fairly egalitarian. From just a surface level view of the vaults it seems like housing is allocated based on the needs of the vault dwellers, everyone participates in the labour required to keep the vault working and everyone alive, and effort is made to place people in careers that make best use of their talents.

This ignores the whole vault-tech evil experiments such as the one mentioned in the show that is understocked to force the residents to compete with one another, but in the "normal" vaults they aren't (to the best of my knowledge) operating on a capitalist mode of society. You don't see a vault dweller worried about making rent, though I won't rule out that it happened