Seattle has year-round farmers markets (here’s a list of what’s available this week) and Pike Place market.
Granville Island in Vancouver is also year-round.
Schenectady has one all year. It moves indoors in the winter.
In the Twin Cities, January and February are usually very thin on produce (though lots of meat, fish, and eggs) but there’s often root veggies and squash around. Early stuff can start showing up in March and brings a premium price.
Can you say “greenhouse”? 
Raleigh, NC has a year roundFarmer’s Market. They are only closed on Christmas as far as I know.
I can. I can say greenhouse.
I need to relocate! 
Chicago has a couple year-round farmer’s markets, and a few neighborhoods do winter indoor markets at least once a month. So I might have to travel out of my neighborhood in the winter, but can still find markets at least once a week. I got lucky though, and have an organic grocer right around the corner who sources a lot of her produce locally, so I can go there every day.
Dallas has some- both the big one downtown, and smaller local ones.
Generally speaking, the produce is pretty seasonal- i.e. you’ll have a lot of tomatoes in June, and a lot of stuff like onions, cabbage and root veggies this time of year. Basically anything that will live through mild freezes will show up this time of year.
NYC’s main market in Union Square is year round and outdoors. It isn’t even sheltered with a roof, like the veggie market in Cleveland. It’s completely outdoors. Actually I’m not aware of any NYC farmer’s market that has a permanent or indoor location. In Ann Arbor the market had a WPA-built roof, no walls, but the veggie market closed in the winter (Artisans all year, IIRC).
Atlanta has a large one open year-round. It’s 150 acres, and open 24/7.
It’s the cheapest place to get pumpkins and Christmas trees that I know of.
It’s kind of a cool place to roam.
I would think that nearly any city of decent size would have a year-round farmers’ market. I live in Kentucky and both Lexington and Louisville have year-round farmers markets. Even the market in my hometown of about 60K people recently became year-round. They do move indoors in winter. Fresh produce available includes greens/lettuce, winter squash, apples, root veggies, eggs, and meat, and processed foods like jams, cheese, pickles, bread, dried fruit, crafts, etc.
Right. The Jean-Talon market gets enclosed from Nov. 1 to (I think) April 30. The rest of the time it is open air. And it is open 7 days a week. Unlike, say, the Union Square Market in NYC, which opens only Tuesdays and Fridays.
The links says the current open days are Monday, Wednesday, Friday, and Saturday.
Baltimore’s 32nd St market is open year-round, rain or shine.
Madison, WI has a year-round farmer’s market that moves indoors during the winter.
Really depends on how you define a “Farmer’s Market”.
One definition might be a market where local farmers sell produce that they harvested from their own local fields the day before or even the morning of the market.
By this definition, the Farmer’s Market in Hilo on the Big Island of Hawaii is open year round selling* fresh* fruits and vegetables.
There are a handful of markets that stay open year-round in Portland (OR), although usually with a reduced schedule.