Cut each bell pepper into 8 strips; cut onion into 8 slices.
Combine vinegar and next 5 ingredients in a shallow dish or large zip-top plastic bag; add bell pepper and onion. Cover or seal, and chill 30 minutes. Add mushroom caps; cover or seal, and chill, turning occasionally, 30 more minutes.
Remove vegetables from marinade, reserving marinade; set mushroom caps aside.
Grill remaining vegetables, covered with grill lid, over medium-high heat (350° to 400°) 10 minutes or until partially charred.
Place 4 mushroom caps upside down, and top evenly with grilled vegetables and cheese slices; place remaining mushroom caps, right side up, over cheese.
Grill, covered with grill lid, over medium-high heat (350° to 400°) until cheese melts, turning once. Serve with reserved marinade.
