Short answer - no.
Long answer - it's not the farmers' market that is organic, it's the farmers AT the market that may or may not be.
Farmers' markets concentrate the farmers and the shoppers at one place and time, making it more efficient for both. The farmers have access to more buyers than they would if they ran their own roadside stand, and shoppers have more variety available. So yes, they ARE your best and most cost effective way of finding and buying bulk produce.
In my area, there are a good half-dozen farmers' markets scattered about the city. They're usually in parking lots, on different days of the week. One is on Tuesday evenings, and blocks off a downtown street. Most run from roughly March to November, based on the growing season. Some of the vendors are organic, some are not. Some vendors don't even have produce - the markets are good places for craftspeople to sell their wares as well. One place that I go to every week is so large that there are sections within it where vendors cluster.
Go to a farmers' market. You'll find greater variety, lower cost, and fresher produce than you would at a retail grocery store. Ask which vendors grow organically. Within a couple visits, you'll develop a list of your preferred vendors, you'll have a pretty good idea of what they carry week to week, and you can even plan ahead, since they'll often tell you what's coming into season.
Enjoy ;)
Steve (curmudgeonly old omnivore)