Soak wood chips in water 30 minutes. Drain well.
Bring the vinegar to a boil in a small saucepan; cook until reduced to 1/4 cup (about 8 minutes).
Combine mushrooms and next 5 ingredients (mushrooms through onion) in a large bowl. Add oil, salt, and pepper; toss well to coat.
Prepare grill. Place wood chips over hot coals. Place mushrooms, squash, zucchini, eggplant, and onion on grill rack coated with cooking spray; grill 5 minutes on each side. Turn vegetables; add tomatoes. Grill 5 minutes or until vegetables are tender.
Arrange 1/2 cup arugula, 3 yellow squash pieces, 3 zucchini pieces, 2 eggplant slices, 2 tomato halves, 1 mushroom cap, and 1 onion wedge on each of 4 plates. Combine the basil, oregano, cilantro, and thyme in a small bowl, and sprinkle evenly over vegetables. Drizzle 1 tablespoon vinegar over each serving.