Yes, you can issue a debit card to an authorized signor of a personal account, provided that doing so is not inconsistent with your bank’s policies and procedures. Since the person already has access to the funds in the account as an authorized signor, the debit card is merely another means for him or her to access the account.
Business accounts can raise different issues, but the answer is basically the same. A corporate customer may want debit cards issued to individuals who are not authorized to sign checks on the account. In other words, authorized signors and debit card holders are not necessarily coextensive.