Answer:
If you don't send the WEB entries to an ACH Operator, but instead process them "as if" they are true ACH items, I don't think that makes your bank an ODFI.
But if they can complete A2A transfers between their accounts at your bank and their accounts at other banks, and your customers initiate these transfers using your bank's account access portal, your bank becomes an ODFI because your bank is submitting those ACH entries via the ACH processor to the other bank. And that applies whether the funds will move to or from your bank. Status as ODFI means you are sending the transaction entry, whether your bank is losing or gaining funds as a result.