Well the errata states that if you would begin the turn with it tapped you may untap it and skip that turn, so technically it begins your turn before an untap step happens.
This is wrong. the "If you would begin..." clause in Time Vault is a replacement effect, which replaces you beginning your turn with untapping Time Vault. You therefore never begin the turn, just as if you replace a draw effect with another effect, you never draw a card.
Just out of curiousity, for what interactions is this relevant?
Oracle text: Time Vault enters the battlefield tapped.
Time Vault doesn't untap during your untap step.
If you would begin your turn while Time Vault is tapped, you may skip that turn instead. If you do, untap Time Vault.
Tap: Take an extra turn after this one.