Vinegar will kill what ever it contacts, grass or "weed".
There is some research that seems to indicate that Dollar Weed can be at least partly controlled by limiting the amount of irrigation because Dollar Weed grows best in wet soils. If you accept the concept those that advocate the use of synthetic materials that you need a "weed" free lawn then the only way to do that is with synthetic poisons. Organic people will properly feed the lawn (not too much), mow the grass they have high to aid in shielding the soil so "weed" seeds have lee opportunity to germinate, and water sparingly. The artifically dark green "weed" free lawns, that are also havens for diseases and insect pests, are not what an organic gardener would be looking for.

Here is a link that might be useful: About Dollar Weed

This weed spreads aggressively by underground rhizomes and is very difficult to get rid of by 'pulling'. As a matter of fact, you might even encourage its spread by breaking off those rhizomes.

There has been some positive chatter about the use of baking soda as an effective post-emergent control of dollar weed. You might want to investigate that. If you have a lot of it, I wonder how useful that would be, however.

You are going to have a very tough time of it.

There is no organic method for treating Dollar Weed. I have tried them all, none work. The best chemical treatment, Image, which works and does not harm southern grasses is suggested in the link given above by Kimm. If you decide to use it be prepared to wait three or four weeks before there is any sign of it working. Those living in the north have no idea what an agressive weed this is. It blankets everything. Sorry that I cannot give you an organic solution. I would love to discover one.

If anyone is going to discover a good organic solution, it will be you, Nandina. ;-) I take it that you've experimented with the baking soda? Darn it; I thought it sounded too good to be true.

I agree with you that Image will do the job on this dastardly weed.

My parents have had great success with the baking soda solution! They live in the Jacksonville, FL area. They wet the dollar weed first (light mist), then sprinkle the baking soda through a fine mesh sifter/strainer. It works overnight- the dollar weed dies, the grass does not. Not bad for spending about a dollar AND being "green!"

Who sez we in the North do not enjoy the battle with dollar weed? I killed mine with vinegar and two feet of leaf mulch before the dollar weed formed new seed pods in a new year.