Vinegar might be an effective "weed" suppressor if repeatedly applied over a long period of time. All that vinegar does is kill the top growth, so the target plant if it has good food reserves in the roots can just grow back unless repeated applications eventually deplete those food reserves.
Now that you have found that the other two, expensive and supposedly very effective "weed killers" actually are not as effective as the ads state, and both of those also require repeated applications, often for many years, you can try a more eco friendly method by laying down newspaper which will, after some time cause the target "weed" to use up the food reserves in the roots and just quit trying to grow. You wil find that just like those expensive "weed killers" you will need to make repeated applications to control this plant, there is no quick solution.

skoot, I don't have much hope for your winning the war on this very persistant weed (Hydrocotyl umbellata) with newspaper layers. I've seen those stolons travel for many feet (yards) under mulch and even plastic just to come up where it can. It would certainly be worth a 'try', though.

I also have little faith in vinegar for this weed. The foliage is simply much too waxy for it to be effective. I've never heard of anyone having any luck with such a treatment.

"Image" is about the only herbicide that has the ability to control this darned stuff. Read up on the product before purchasing, then read and follow the directions to a 'T' if you do decide to try it.