Most smaller banks now have outsourced their holdings to larger banks anyway so chances are your account info is in a major data center and accessable somewhere.
My wife worked in HR and during Katrina they were still direct depositing payroll into peoples accounts so they could continue to withdraw or use a debit card somewhere to get what they need in the town where they evac'ed to. The people with mailed paychecks were in trouble because they couldn't provide addresses or proper identification saying they were who they were in the new location they were at.
Its still best to keep cash on hand, I've managed to save up a little bit in my EDC just for those situations.