I'm having a problem with the company that issued a gift card. Can the OCC help me?

Maybe, depending on who issued the gift card. The OCC has authority over national banks. If a national bank issued the card, it would be appropriate to contact the OCC with your concerns. But if someone else, such as a retailer, issued the card, it would be more appropriate to contact the attorney general's office or another agency in your state with responsibilities for consumer protection.