@Narsi - If you happen to pass by Annanagar East, you can try this place - The Green Shop, they are located right opposite to the Karaikudi Restaurant #9 B Block, 2ND Avenue, Anna Nagar East, Chennai - 600102.. Am not sure if they are in Nungambakkam. But I believe that they have a store in Kilpauk. Also found some interesting reviews about this place - http://www.asklaila.com/listing/Chennai/Anna Nagar/The Green Shop/T03gZniv/
Dhanyam Organic Superstore- N Boag Rd, T Nagar, Chennai - 044 28157654. They seem to sell organic vegetables and fruits, spices, dairy products, health supplements, baby foods etc... Website: http://dhanyam.in/
God! I'm still tryin to recollect the name of the shop! Its located on the second floor of express avenue mall, next to max lifestyle. they sell organis spices, organic food. they dont deal with organic vegetables though
You might want to try Brown Tree , a Organic Food Departmental Store near Hotel Apoorva Sangeetha, in Nungambakkam High Road. I noticed a board outside the store which said that they sell dry fruits, spices, sugar- free products, health food , organic food etc..