No, they will not accept a picture of your gift card. You can store the gc on the disney gift card website but all that lets you do is transfer the balance to another gc, you can't spend from the website or see the card data from the website (and if your Disney Gift Card account is hacked someone can transfer your balance to their own gift card).
Better idea- just charge your food direct to the gc instead of to your room. That way you don't worry about losing the gc before the very end of your trip when you're settling your hotel bill.