Is a power of attorney allowed to have an ATM card?

Generally speaking, if person A gives a power of attorney to person B, then person B can attend to any or all of person A's business, including banking, in the same manner as person A could do himself or herself, subject to any limitations contained in the power of attorney document itself. For specific legal advice, consult a lawyer in the jurisdiction where the power of attorney was given, preferrably with a copy of the document itself.