Am I allowed to have 5 Bottomless Vaults in my deck?
No, the maximum allowed number of copies for each card is four. The only exceptions are basic lands (Forest, Snow-covered Forest) and cards which specifically mention they're exceptions (like Relentless Rats).
100.2a In constructed play (a way of playing in which each player creates their own deck ahead of time), each deck must contain at least sixty cards. A constructed deck may contain any number of basic land cards and no more than four of any card with a particular English name other than basic land cards.
(from the Comprehensive Rules)
So in the rare case that you're playing a Fifth Edition draft (that's a Limited format) and manage to draft five Bottomless Vaults, then it would be allowed.