When it comes to food, I feel pretty lucky here in Beirut. Fresh local produce year round (thank you temperate climate!), a culture that favors a vegetable-heavy cuisine, and very reasonable prices at the market are all fantastic aspects of life in Beirut.
And, if you're into organic produce, you're in luck.
There are the farmers markets that take place on various days around town and if you can't make it to one of those, you can find certified organics in the big grocery stores.
And, there's Healthy Basket, a shop in Ras Beirut that will deliver a weekly basket of organic "stuff" to your door--I say "stuff" because in addition to the produce that fills most of the basket, you might also find eggs, grains, etc. thrown in there too. Here's a link to their contact info if you want to sign up for a weekly basket of your own.