If you are looking for a selective weed killer that is certified organic I don't think you will find one. There are organic products that kill vegetation, but if you wish to kill weeds growing in a lawn or amongst desirable vegetation, I know of no such product.

People have used corn gluten meal as a germination inhibitor with mixed results, but that can only take care of annual weeds by preventing the overwintered seeds from sprouting, the perennial weeds will not be harmed. I don't know what the status of corn is with the standard's board.

There are certified organic herbicides....they tend to be plant acids (citric, acetic) and essential oils. Also a couple based on alpha terpineol. But as noted, they are not selective but broad spectrum products and have the ability (not sure how effective) of interrupting the growth of any plant to which they are applied. Most require specific temperatures and sunlight to be effective, which obviously limits their efficiency. And they are not inexpensive.

A couple of brand names you could look for are AllDown, SmartDown and Nature's Avenger Weed Killer. Be sure to study dilution rates......as with acetic acid (vinegar), the higher the concentration, the better the results.

If you absolutely must spray something, start with household vinegar at 5% acidity. Spray at the hottest time of the hottest days. Give it three days to show results. If that isn't fast enough for you, look for vinegar at 20% acidity. This stuff is pretty hazardous if you get it on your skin and NEVER get it into your eyes. But it will knock many weeds down to dust (not a metaphor) in less than an hour. If you have to have a vinegar that is labeled for organic horticultural use, you may have trouble finding it. Lady Bug used to have one and may still have it on the market. There was some moaning by the chemical companies that vinegar had never been proved to work and so could not carry a herbicide label, but that should have been put to bed by now. Still the warning label on high strength vinegar is woefully inadequate. You can go blind for weeks if you splash some into your eyes.

Do you have a Weed Hound tool? Of all the millions of words of advice on Gardenweb, the best I've ever seen was recommendations for that tool. Do a search or just go get one.

Someone should probably ask what weeds you are worried about. There are 'non crop' plants which are beneficial as indicators of other problems or as breeding hosts for insect predators. There is also something to be said for taking the live-and-let-live approach. Orchard growers in South Texas have learned to let nutgrass grow wild. First of all it doesn't take much from the soil, it provides a natural shade/mulch to keep the soil cool (less evaporation than from bare soil), and not spraying herbicide and worrying about it boosts their profit.

Where do I get 20% acidity Vinegar?