How can one test if a vegetable is organic?
No, there is no (reasonable) way to check it for yourself. 

There are thousands of toxins in the world, and any of them could land on your vegetables. Some of them might be easily discovered, but most of them would require a complicated lab test. Even if you had the knowledge required to conduct such a test, and the equipment needed, you would have to run hundreds of tests on each batch of veggies. There are attempts to develop easy tests (full text behind paywall) for consumer use, which cover a few common pesticides, but I don't think they have been turned to a commercial product yet. 

This is why there are regulations, and government organs which enforce these regulations. Sure, they don't have the capacity to test each tomato field for pesticides. Their control is based on levels of risk - for example, each slaughtered pig should be tested for parasites dangerous for humans, but for organic vegetables, they are only taking random samples. It is expected that the risk of fines and other financial loss will deter farmers from using pesticides on vegetables labelled as organic. Additionally, independent eco organisations run their own tests and expose producers who sell contaminated vegetables. This results in a functioning system, where only a small fraction of organic food is actually contaminated. 

If you have reasons to suspect the vegetables you aren't in fact organic, you could always pay a lab to run tests for you. It would probably cost a lot more than the price of the vegetables, so I can't imagine a situation where this would make sense.