There is a Walmart Super Center off Hwy 192 just west of World Drive. I went there several times during our stay last November (week b4 Tgving, week of Tgiving). Walmart Supercenter, 2855 N Old Lake Wilson Rd, Kissimmee, FL 34747
Personally I liked hitting it in the early morning as it gets crowded during the daytime. But I went there all times of day over 2 weeks. Our mode is to bring a plug-in cooler from home with a little bit of food but we bring all the condiments etc and food/drink for the first Fort night. I like to do the first main shopping after I set up at the Fort. But if you want to shop before you arrive, this is the closest. You would come down the Turnpike, turn south on Hwy 429 (toll), get off at Hwy 192, and head in that way. Just stay in right lane leaving WM, right lane on 192, and curl up onto World Drive.
If you have to head back there during your stay, I loved cruising down World Drive and merging smoothly west onto 192. But what was frustrating was it would take 2-3 light changes to make that left turn off 192. The easiest path from the Fort to the store (for me) was go out Osceola Parkway past the AK parking lot, turn left at the traffic light, and curl back onto Sherberth Rd and take it STRAIGHT ACROSS 192 and then get over to WM.
They also have a Murphy gas station there FWIW which had best pricing around the Mouse.
We stayed till the Monday or Tuesday after Tgiving (Preferred loop 100, site 14 next to the paved walking trail to the Settlement TP). IMO there certainly were departures that Fri-Sat-Sun after Tgiving but new people coming in too. The Fort has become more popular for those first 2 weeks of December (and the pricing reflects that) and it's not a bargain or half empty any more during that period. My advice, if leaving the day after Tgiving, would be to start the return journey as early as possible in the morning. The Turnpike is not a problem but when you reach I75 you get all the travelers coming up from Tampa and it can get very crowded (towing is no fun then) and fenders get bumped, people pull over, and it's a 10mph crawl for 2 hrs to get up to the scene and see everyone is okay. Once you get up to I10 it's smooth sailing. But the earlier you can start, the fewer fellow travelers on the road.
Bama Ed
PS - lots of DIS folks are at the Fort in November so we usually have a group "November Trip Report" thread that people can post into. Last year's thread is here:
https://www.disboards.com/threads/november-trip-reports.3899929/
Our trip starts on page 3 post #52 if you want to see what goes on then.