As far as supermarkets are concerned, here's a list of those closest to the Strip:
There's also a branch of 99 Ranch Market, the Asian specialty supermarket chain, in Las Vegas' Chinatown, located at W. Spring Mountain and Wynn Road, where you can purchase myriad fresh fruit and veggies.
Unfortunately, acclaimed Chef Mario Batali's "Bet on the Farm" Molto Vegas Farmers Market, which was located on Dean Martin Drive, closed in February of this year for lack of a business license, but there are now almost daily farmers markets across the Vegas Valley (the one downtown is closed for the winter season), if you have transportation and are looking for local produce.
If you visit the Las Vegas Farmers Market website, you'll find details of the weekly events at Garden Park (west side) on Tuesdays, Bruce Trent Park (also on the west) on Wednesdays, and Floyd Lamb Park at Tule Springs (first and third Saturday of the month).
STOP THE PRESSES! Having written this answer, we just read on Twitter that the Bet on the Farm farmers market is back, commencing this Thursday at Springs Preserve from 10 a.m. to 1 p.m. Great news! Click here for more information.