The Vault Police feature into the start of the film prominently, arresting the protagonist for 'disturbing the peace' after he rants about how manufactured life in the vault is on the metro. Although, Female Officer is the only known member of the Police, it is used in a plural context in its first mention, so it is likely that more than one officer was dispatched to arrest him.
They use video surveillance of the vault in order to dispatch officers when and where it is appropriate. It is not known if they also patrolled the vault, but it seems likely in light of other vaults' security teams' procedures.
The only known rank, is that of 'Officer'. However, it seems that they answer directly to the elders (as a group). Another interpretation of this is that the Vault Police are in the pocket of the Vault Supervisor. This is suggested in that the Police are forced to let Hero off, but later do nothing to protect the Overseer, instead presumably siding with the greater majority of the elders and populace.