First of all, there the Doctor can only get into the vault during the solar storm for one of the following reasons, likely both:
- He needs the solar storm to damage the lock to the vault.
- If he finishes his job too early, he cannot properly initiate the time loop by giving Madame Karabraxos the paper with his number at the right time.
Thus he needs to smoothly break into the bank at the right time.
why not get into the places he could infiltrate before the storm and then wait for the storm to hit before breaking into the vault?
Risk of detection: Not only is there the teller, but they could have all sorts of security mechanisms such as flooding the entire facility with lethal radiation from time to time. It is, after all, the safest bank in the universe. A case is much less vulnerable and less easy to detect than the Doctor.
if he can infiltrate the bank to do that, why not just do the whole operation on his own rather than recruiting the team as he did?
We actually never see that the Doctor was completely on his own when depositing the cases. Thus the placement of every case could have been a similarly complicated undertaking as what we are shown in the episode.
Moreover, all of the cases seem to be located in a position where they could stay for at least some time (let’s say a few days) without being detected – in particular if some security systems that could detect them are bypassed (e.g., cases are positioned outside the view of security cameras; cases are equipped with some cloaking device that switches off at the right time; some help from the old Madame Karabraxos):
- The first case has be deposited in the vault itself, which could have happened years before.
- The second case is on some remote platform within the support structures of the bank, which does not look as if people would get there often.
- The third case is somewhere at the access terminal to the vault (we do not see where it is located – the scene cuts to the Doctor holding it in his hands). While an important location, people do not seem to go there very often either, judging by the lack of decoration.
So, the Doctor could have positioned those cases at whatever time is suitable for doing so, e.g. when the bank is closed. In contrast, when the actual heist happens, he does not have the luxury of being able to choose the right moment and thus needs the cases (because otherwise the teller could tell or security mechanisms would find them).