You can find a few things in the gift shop in GCH; they actually have some milk, cereal and cold drinks. You can also get a larger variety in WhiteWater Snacks, also in GCH lobby. Expensive, though, of course.
There's a shop in Tropicana, there's also one inside HoJo. There's a 7-11 on the corner of Harbor and Katella.
I don't know about "full-blown grocery store" but we were in the same situation when we stayed at GCH in March. What I really wanted, though, was a rack of bottled water, because the tap water is truly horrible, and bottled water averages $3.00 a bottle in the hotels and parks!

Then we needed breakfast stuff. So we ordered online from Albertson's for delivery. It worked great. They don't have a minimum amount you have to order. We got yogurt, cereal, milk, juice, pop tarts, apples and bananas, a roll of paper towels and a package of plastic spoons. Plus the aforementioned 24-pack of bottled water. They delivered to the hotel about a half-hour before we arrived, and the bell desk held it for us.