A & P Food Store
701 Royal St, New Orleans, LA
Visit the tiniest supermarket in the city

Matassa's Market
1001 Dauphine St, New Orleans, LA
A French Quarter favorite

Royal Street Grocery & Deli
801 Royal St, New Orleans, LA
Deep in the French Quarter with the latest opening time so far--10pm.

Quarter Grocery
836 Burgundy St, New Orleans, LA
Another neighborhood market in the Vieux Carre.

Verti Marte
1201 Royal St, New Orleans, LA
This corner store is best known for its late-night po'boy kitchen, which is currently closed, but the market section is open.

They also have the french market where you can get fruit, etc.