My husband and I stayed last year in Villa Thalassa which was right on the beach.  We booked through Agni and there is a grocery in Nissaki called Aphrodites that will deliver.  I worked with Agni to give them a list, unfortunately my email never got through to them, which we didn't realize until we actually got to our Villa.  Aphrodites will also pick you up to get groceries, it isn't huge but it was big enough. 

For the most part we used the little corner store ran by George, now this is close to the beach, so depending how high up on the hill you are this may be a hike to take your groceries back.  He had the basics, fresh fruit and fresh baked bread every morning.  Beer, water, eggs, soda, bacon and other essentials. George even had beach shoes and snorkels, both of which we were very grateful for! We arrived late in the day, and the little corner store was closed, but George was sitting in the Taverna by the Beach and he opened up the store and let us get some essentials until we made it to the grocery.  

Really the most cooking we did was breakfast, we ate out for Dinner.  Highly recommend Spiros Taverna on the Beach great food and service and really good prices.

Last year was our first trip to Corfu too, and we had a wonderful time.  Hope you enjoy!