To prepare vinaigrette, combine first 6 ingredients in a small bowl; stir well with a whisk.
To prepare vegetables, combine 1/4 cup vinaigrette, zucchini, and bell peppers in a large bowl. Brush remaining 1/4 cup vinaigrette over both sides of onion slices and mushroom caps. Place vegetables on grill rack coated with cooking spray; grill 2 1/2 minutes on each side or until vegetables are tender and browned. Place vegetables on a cutting board; cut each vegetable piece in half. Serve over rice; sprinkle with cheese.