Mix oil, vinegar, Steak Seasoning and Italian seasoning in small bowl. Reserve 1/3 cup for brushing on rolls. Place steak in large resealable plastic bag or glass dish. Add remaining marinade; turn to coat.
Refrigerate 30 minutes or longer for extra flavor. Remove steak from marinade. Discard any remaining marinade.
Grill steak over medium heat 6 to 7 minutes per side or until desired doneness. Split rolls in half. Brush with reserved marinade. Toast rolls on the grill, open-side down, about 30 seconds.
Slice steak into thin slices. Serve steak on toasted rolls topped with cheese, lettuce, tomato and red onion.