I am not so sure for Neighborhood Markets. Those are like grocery stores and are usually not stand-alone stores which have control over the parking lot. They usually have other stores in the same strip mall area. Of course the Supercenters control the parking lot and they usually allow overnight parking. You should check with the store manager when you arrive to get the lowdown on if and where you can park.
Yes, any Walmart is safe haven for homeless as well
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