Combine first 7 ingredients in a large bowl. Set aside.
Cut carrots and remaining vegetables into large pieces.
Add vegetables to vinegar mixture, tossing to coat. Let stand 30 minutes, stirring occasionally. Drain vegetables, reserving vinegar mixture. Arrange vegetables in a grill basket.
Cook, covered with grill lid, over medium-hot coals (350° to 400°) 15 to 20 minutes, turning occasionally.
Return vegetables to reserved vinegar mixture, tossing gently. Serve warm or chilled.