Prepare the marinade. In a medium bowl, whisk together the olive oil, balsamic vinegar, lemon juice, garlic powder, dried basil, salt, and black pepper until well combined. Reserve half of the marinade in a separate bowl to drizzle over the vegetables just before serving.
Prepare the vegetables. Wash and dry all of the vegetables. Peel the carrots if desired. Slice the zucchini and yellow squash into ½-inch rounds. Quarter the red onion, leaving the root end intact so the wedges hold together. Slice the bell pepper into wide strips and leave the cherry tomatoes whole.
Marinate the vegetables. Place the prepared vegetables in a large bowl or gallon-size zip-top bag. Pour the remaining marinade over the vegetables and gently toss until everything is evenly coated. Cover or seal and refrigerate for 1½ hours, tossing once or twice during the marinating time if possible.
Preheat the grill. Heat an outdoor grill to medium heat, about 375 to 400 degrees F. Lightly oil a grill tray or grill basket if needed to help prevent sticking.
Arrange the vegetables. Spread the marinated vegetables in a single layer on the grill tray or grill basket. Avoid overcrowding so the vegetables cook evenly and develop light grill marks.
Grill. Place the grill tray over direct medium heat and cook for 10 to 12 minutes, turning the vegetables halfway through with tongs or a spatula. Cook until the vegetables are tender with lightly charred edges. Cherry tomatoes should be softened but still hold their shape.
Transfer to a platter. Carefully move the grilled vegetables to a serving platter, arranging them by color or grouping similar vegetables together for a beautiful presentation.
Finish and serve. Drizzle the reserved marinade over the warm vegetables just before serving. Garnish with chopped fresh parsley, basil, or grated Parmesan cheese if desired, and serve immediately alongside grilled chicken, steak, burgers, pork, or seafood.