No, it can't be done. I'd suggest something like a earthbox or similar self-watering container for organic growing. There's plenty of information at the site below.

Here is a link that might be useful: Gardenweb container forum

I had the same question in my mind before. When I saw a demo farm that looked like "hydroponics", complete with pvc's, pots, media, etc., where feeding is done only 3 times a day for ten minutes each, I wondered. Well, what do you call that. In the final analysis, it looks like a marriage between hydro and organic methods. The solution they use is an ordinary soil type fertilizer. Another observer says it's not hydroponic but plain and simple sub-irrigation, an automatic watering system. Many of the drip and flood/drain system may be used either way. Pure hydroponics (according to Resh) is a set-up where roots are directly in the nutrient solution. The only function of the medium is to hold the plant in place, nothing else. The 11 planter may be used either as a hydroponic system or sub-irrigation set-up. Make your choice.

Plants do not need microbes or bacteria as long you provide the mineral nutrients they require. Plants can be grown under sterile conditions and do fine.

Even hydroponic systems that use no solid medium will almost cetainly have some microbes in the nutrient solution, living off organic materials secreted by the roots and pieces of dead roots. You would have to go to great lengths to provide a completely microbe-free hydroponic system.

Here is a message by DR Ingham of the Soil Food Web Oregon which was featured in the Growing Edge magazine. It concerns the use of compost tea with hydroponics for good microbes. A very good read. Link:
http://www.soilfoodweb.com/sfi_html/news/compost%20tea%20and%20hydroponics.html

I've checked out a book from the local library called Grow More Nutritious Vegetables Without Soil: New Organic Method of Hydroponics C1983. The author believs there are certain microbes that plants can take up which could make vegetables more nutritious. I don't recommend the book becouse he says his method is not proven to be more nutritious and his method in the book is not as he claims in the title "organic". He mixes commercial inorganic formulas with commercial organic formulas.

Here is a link that might be useful: Compost Tea and Hydroponics

There are bacteria and fungi additives that are organic (OMRI listed). They act as incredible growth promotors. The techncial info and list of organisms can be found at www.GrowMoreRice.com. The site is targeted towards rice, but the products work on any plant.

Here is a link that might be useful: Biological Soil Amendments - bacteria and fungi