We use a Publix that is a little out of the way, but it is fairly new, very clean, and easy to get to.
You head toward the Animal Kingdom Lodge, staying in the left lane. There is a left hand turn that will take you off Disney property just as you approach the lodge (this is past the turn off for the park). Stay on this road until you almost reach 192. Just before 192 make a right hand turn onto Black Lake Road (this road runs parallel to 192). Follow this road all the way to the end (it ends at a light on 192). Turn right and there is a little plaza with a Publix right down the road. It's on the right hand side.
It's probably not the closest Publix, but we really like it, and it's very easy to get to.