aa.com's shopping portal allows you to earn miles with on-line purchases at Home Depot. You can get miles when redeeming a gift card, but you cannot get miles when buying one. Though, IIRC, last year they ran a special for a short time allowing miles when buying gift cards.
Some sites will allow you to use a gift card and another form of payment, i.e. CC, at the same time. This is important in case your purchase is for more $$ than are on the gift card. Physical Home Depot gift cards are bar-coded, and they keep the balance on a server. So if after using it at the shopping portal you have a little bit remaining, you can use it later either online or at a store.

Be aware that some online merchants do not give miles when you are using gift cards for the purchase. Read the exculsions always. And read them every time because rules change. E.g., in 2006 Home Depot changed its rules. Before then they gave miles when purchasing gift cards and it was a wonderful double-dip opportunity, not to mention the ability to get miles for in-store purchases. I got miles for my new deck!