I'm in a slightly similar situation; Delaware C-corp, but we don't do business there. Our bank account is in NYC with Citibank, merchant account with First Data. We are an online SaaS business.
IANAL etc but I thought it was best not to do any business in Delaware directly, so not sure why/if it matters which banks in Delaware would deal with you. In any event, I think you really just need an EIN to open a bank account.
The whole subject of where an Internet business has "nexus" is apparently quite a complicated one.
Not sure if the above is of any help, hopefully it is.