In every store, some hotels, and the department store there will be an ATM. It uses your ATM card which is in your pack by default.
You won't get money deposited for battles until you talk to your dad on the phone, so make sure to save a dollar for payphones whenever possible.
You have to have the ATM card with you to withdraw, so if you stow it in storage you will need to take it out again to withdraw.
Because storing it for the item slot is very slow and cumbersome, I recommend just keeping it on Ness, as his magic and high base stats cover for what that item slot could have been.