Chop all the vegetables, then add them to a large bowl except for cherry tomatoes.
Preheat grill to high heat, about 400 to 450° Fahrenheit.
In small bowl, whisk together oil, soy sauce, Dijon mustard, garlic and onion powders, salt and pepper until well incorporated.
Add marinade mixture to veggies and mix well until well coated. Let sit for about 10 minutes.
Place cherry tomatoes into empty small bowl where marinade was, and mix to coat with leftover mixture. Set aside.
While veggies are marinating, place empty grill basket on top of grill grates to heat up, about 5 minutes.
Add veggies (except tomatoes) to grill basket, and cook for about 10 to 13 minutes, stirring with wooden spoon or tongs often.
After 10-13 minutes, add cherry tomatoes to rest for veggies, mix them in, and continue cooking for another 5 to 7 minutes or until veggies are tender when poked with tines of fork and tomatoes are just starting to burst. Once veggies are cooked through, carefully remove hot grill basket from grill, transfer them to bowl or plate, and serve warm with sides of choice.