Renting the car for a day would also be one of my thoughts if what you are talking about is things you normally get in a supermarket.
Nevertheless you do have the option to ship and you can pick any reasonable time before the trip such as a regular type delivery that takes several days with the plan to have it arrive the day before or of your arrival. I would not do it as an overnight delivery because it is a lot cheaper to do one that takes two days or even five. You send it to yourself as a guest c/o the resort, date of arrival and if it arrives early they just hold until you get there. You have a package charge as mentioned above if you are staying at BCV, BWV or BLT.
Do not send perishables. Also, I would avoid things in glass that could break. Anything such as liquids in plastic, toothpaste, similar should be new and unopened to minimize risk of opening and spilling over everything else in the box. Otherwise, what you send may depend on size of room you have since you can do less in a studio than larger; things to consider particularly if getting 1BR or larger (but many can also apply to a studio) include: (a) laundry necessities (detergent, softener, spot removal); (b) bathroom necessities (soaps, shampoo, conditioners, skin creams, toothpaste, TP, etc.; (c) packaged or canned goods such such as cereal, Pop Tarts, possibly bread, even bags of chips, canned soup, canned pop, kool-aid, plastic bottle water, coffee, sugar, coffee filters, mustard and catsup and salt and pepper (if desired), baggies, plastic trash bags, (c) particularly if you are setting up an Owners locker, you can send beach towels (if you prefer them to resort towels for pool), sandals or flip-flops you may want to leave there, likewise any clothing, jackets, that you might want to keep there for continuous use, any DVDs you would like to have and keep there, hats you may only use at Disney.