CLOSEST supermarket to Downtown Disney, though (and so to POR) would be Goodings, at the Crossroads Shopping Center at the opposite end of Hotel Plaza Boulevard from DTD. It's about a mile, if the OP wanted to walk, but that does involve crossing a couple of wide, BUSY streets.
But a cab from DTD to Goodings would be much less than $13, and back to POR would cost less than $24.
Yes, I realize Goodings' prices are higher than Super
Walmart - but balance that against the difference in transportation prices. The OP would have to be buying a LOT of groceries to make a cab ride to SW cost-effective.
In fact, for $17 service charge,
www.wegoshop will shop at the OP's choice of one store and deliver the groceries to POR; $12 on a $50 minimum order,
www.gardengrocer.com will do the same, but from a predetermined store with predetermined prices.
Both companies get good reviews on even subjective items like produce, and either service charge is cheaper than the total cab fare would be.