There is actually a chance that the vault will be open when you start the heist. In this instance, you do not need any keycards to get into the vault and can just begin the heist.
If the vault is closed, you will need two key cards to get into it. These key cards can spawn in any of the following places:
- The street in front of the bank
- The carpark behind the bank
- The bank manager
- The assistant bank manager at the front desk
- In the briefcase next to the cashiers
The cars are simply a case of walking up to the cars that allow you to interact with them and opening them up. It's normally only the back of the car you're able to open up.
The Bank Manager will wander around - sometimes he'll walk out back and sometimes he'll walk out front onto the street. If you get him when he walks out back (since the front of the bank is all glass so it's easy for the civilians to see you) you'll have no issues dealing with him.
The assistant bank manager however stands alongside the clerks and he doesn't seem to move. If you don't have both key cards by this point, you'll need to be taking control of the lobby. Since the vault is visible from the lobby you're going to have to take the lobby anyway.