Heat oven to 375°F. With vegetable peeler, remove all of outer orange portion (zest) in one long piece from oranges. Cut oranges in half. Squeeze juice from 1 1/2 oranges to measure about 1/2 cup. Stir oil into juice.
Sprinkle chicken, inside and out, with salt and pepper. Place orange zest from 1 orange and remaining orange half inside chicken. Tie legs together with string. Insert meat thermometer so tip does not touch bone. In 13x9-inch (3-quart) glass baking dish, pour 1/2 cup water; place chicken in dish. Pour orange juice-oil mixture over chicken.
Roast uncovered 1 hour 45 minutes, spooning drippings from dish over chicken every 30 minutes, until thermometer reads 180°F and legs move easily.
Cut remaining orange zest into thin (1 1/2x1/8-inch) strips to make about 1/4 cup. In 2-quart saucepan, place orange strips and 2 cups water. Heat to boiling. Drain and set orange strips aside. In same saucepan, heat sugar and vinegar over high heat to boiling. Reduce heat to medium; cook 5 minutes, stirring occasionally, until slightly thickened. Remove from heat; stir in orange strips and 1 cup orange juice.
When chicken is done, place in another shallow pan; cover to keep warm. Into small bowl, strain drippings from dish; skim off and discard fat. Add to mixture in saucepan. Cook over medium heat 20 to 25 minutes, stirring occasionally, until slightly thickened. Serve sauce with chicken.