CSA's or Green Carts?
I have a 'market CSA' account with Norwich Meadows Farm.
They are at Union Square Sat. , Tompkins Square Sun and other locations.
Go and speak to Zaid or Haifa about putting money 'on account'. If you apply early in the year, you can get a substantial discount -- but you get your pick of what's available, including eggs, chicken, etc. I love CSA but it's just not convenient for me-- this is a much better arrangement, and I go every weekend.
a lot of those random fruit/veg carts on various street corners everywhere (catering mostly to workers in the area) get their veg trucked/shipped in from anywhere, really; no local credentials at all; just your typical produce that gets sold in markets/bodegas/chinatowns all over the city.
anyway, try this, might help:
I think they source from Manhattan Fruit Exchange. Though, Im not sure why I think that.
The farmer's markets around town will give you a guaranteed local source. Union Square is open Mon, Wed, Fri, Sat. Other nabes have markets at other times/days.