I am the queen of gift carding, thanks to the Scrip system of fundraising.
No, you cannot use a gift card to purchase another gift card
This is not totally true. You can use a store gift card to buy another store gift card. But you can not use a store credit card to buy most of the prepaid visas or mastercards, but I have used them to buy American Express cards.
I do this all the time- I recharge my Kroger card and buy gift cards to the stores not on the list from Great Lake Scrip. I still get the 4% credited back to my school account because I paid for it using the Kroger card.
If your school is not using the Great Lake Scrip gift card center, you should suggest it. I know PS around here use it for bands, choirs and sports.