What flavors, when added to the steaming water can impart its flavor to vegetables being steamed? I know ginger has some water soluble compounds. What else is similar?
Water solubility isn't enough - that gets the flavour into the water but says nothing about whether the flavour gets to the food. Consider salt as an extreme example.
You may get somewhere by putting the source of flavour on top of the food being cooked. There are recipes that do this with mint for example, though they normally add more mint at the end.
The ginger you mentioned may work in a similar way, but testing is difficult as this approach tends to scent the kitchen or even the whole house, so you perceive the flavour even before the food is served.
Perhaps I can suggest a different cooking method. I have recently tried and grown fond of frying our veggies. Yes...frying! Use a medium heat with a nonstick pan and some oil(I use avocado oil). I use a lid for a couple of minutes so the moisture in the veggies helps cook as it becomes steam. The last couple of minutes, I remove the lid and add seasoning. The seasoning can be salt, spices or a combination of. Soya sauce, oyster sauce, worechester sauce . Whatever you want to try. Never any veggie leftovers anymore!