Combine minced garlic and chopped rosemary. Remove and discard giblets and neck from chicken. Rinse chicken with cold water; pat dry. Trim excess fat. Starting at neck cavity, loosen skin from breast and drumsticks by inserting fingers, gently pushing between skin and meat. Rub garlic mixture under loosened skin over breast and drumsticks. Place 5 garlic cloves and rosemary sprigs into the body cavity. Place chicken, breast side down, in an electric slow cooker. Cover with lid; cook on high heat for 1 hour. Reduce heat setting to low; cook 7 hours. Remove chicken from slow cooker, reserving drippings. Discard skin from chicken.
Place a zip-top plastic bag inside a 2-cup glass measure. Pour drippings into bag; let stand 10 minutes (fat will rise to the top). Seal bag; carefully snip off 1 bottom corner of bag. Drain drippings into a small saucepan, stopping before fat layer reaches opening; discard fat. Add orange juice and vinegar to saucepan; bring to a boil. Reduce heat; simmer 10 minutes.