Remove mushroom stems, and discard; cut mushrooms in half. Diagonally slice squash and zucchini; cut bell peppers and onion into 1-inch strips. Place vegetables in a large zip-top plastic freezer bag.
Whisk together oil and next 5 ingredients until blended. Pour over vegetables. Chill at least 30 minutes. Remove vegetables from marinade, reserving marinade.
Grill vegetables in a grill wok or metal basket, covered with grill lid, on medium-high heat (350° to 400°) 10 minutes, stirring occasionally and basting with reserved marinade.