[strike]What[/strike] Do we need to say "Who do you bank with?" or "Whom do you bank with?"?
Note my corrections above. "What do we need to say?" is a standalone question. You can't follow it with a continuation of the sentence/question unless it's something like "What do we need to say if we want to ask someone for directions?" (that's just an example).
While you are still having problems with punctuation, I think you should lay out your questions differently, especially when you're asking us to choose between two options. I suggest that you phrase your questions like this in future:
Which of these two sentences/questions is correct?
1. Who do you bank with?
2. Whom do you bank with?