Answer:
For debit cards, if it's going to replace the existing debit card, provisions in Reg E at 12 CFR 1005(a)(2) give you that authorization. If it is in addition to the existing debit card, provisions in 1005.5(b) and comment #5 in the Official Staff Commentary to that section say you can do so, but such card must be issued as not validated and the institution must follow specific disclosure and validation processes further spelled out in that section.