You could also use Disney transportation to change resorts and have groceries delivered.
That's what I would do..use either gardengrocer.com or wegoshop.com
They will deliver, for a small fee (around $12-15). They bring it right to the resort. Then, have Disney move the bags, but be prepared to not have them available until late afternoon.
I usually hand over my bags before 8am, then leave the resort and head to a park. After lunchtime, I head over to the new resort, get checked in. Sometimes I'm lucky and can get right into a room. Later on that day, when my bags get to me, I can unpack. If I have a grocery delivery, I have them deliver it late afternoon..hopefully I'm in the room at that time and can get all settled.
But, if you would rather use a car service, check with them and find out about grocery stops. For your needs, the Hess Station, over by BW, may fit what you need easily. And it won't be way out of the way..making the driver happier. It should cost under $50 for a car service..which would be my choice between a cab and a car service.