Vinegar will kill tender foliage on anything it hits. Depending on how established the Johnson grass is, you might only kill your veggies and not the grass. Digging out Johnson grass is still often the best approach, especially if it is mature.
There are some products like Avenger (or you can make your own) that
might challenge the robust Johnson grass. It is a vinegar and orange oil combination (here is the
vinegar herbicide recipe from the Natural Organic Library). According to the
Johnsongrass entry, the grass in some areas is now unresponsive to glyphosate (aka Roundup), a product that is never recommended but some might be tempted to try.