If you use your DL credit card for any purchases, it doesn't matter who it's for. Just like if you used any other credit card to make a purchase for someone else. You would get the miles plus double miles if making Delta purchases. If you wanted someone to be able to make purchases using your CC, then you coudl also add them as an authorized user on your account. They would get their own CC in their name, but all purchases would be credited to your account. You would also be responsible for paying off any purchases that the authorized user makes, whether you approve of them or not.
Edit: just realized this is referring to a credit and not a credit card. oops!