Hi Narelle:
There is no real grocery store located on Walt Disney World property, but there are a few grocery stores in the area. There is a Publix about a 20 minute drive away. If you don't have a car with you, you will need to take a taxi. However, personally speaking, if you don't have a car, I would not spend the money on taxi fare (nor spend vacation time at the grocery store). If you decide to take a taxi, your taxi driver or resort will almost certainly know how to get to Publix or a different market.
There are a couple of other options to consider. First, depending on what items you may need, you may be able to satisfy your food shopping needs on property. Port Orleans has a mini-convenience store where you can find a small selection of basic food items such as milk, cereal, eggs, canned goods, juice, bread, soda, etc., but the food selection is limited.
Another option is to arrange for a grocery delivery to your resort. Check out gardengrocer.com, an online grocery store that will deliver your order directly to your resort. You can even place an order before you leave for Disney World and arrange for the food to be delivered after you arrive at the resort.
Enjoy your vacation and happy shopping.