1. Combine 2 tablespoons vinegar, 1 teaspoon oil, garlic, and oregano in a large zip-top plastic bag. Add steak to bag; seal. Let stand at room temperature 20 minutes, turning once.
2. Heat a grill pan over medium-high heat. Remove steak from bag; discard marinade. Sprinkle 3/8 teaspoon salt and 1/4 teaspoon pepper evenly over both sides of steak. Coat pan with cooking spray. Place steak in pan; grill 5 minutes on each side or until desired degree of doneness. Remove steak from pan; let stand 10 minutes. Cut steak across the grain into thin slices.
3. Combine remaining 1 tablespoon vinegar, 1 tablespoon oil, 1/8 teaspoon salt, 1/8 teaspoon pepper, yogurt, and mustard in a bowl, stirring with a whisk.
4. Cut off one-third of each pita; discard or reserve for another use. Arrange 1 lettuce leaf, 1 onion slice, one-fourth of cucumber slices, 2 tomato slices, and one-fourth of steak in each pita; top each sandwich with 1 tablespoon yogurt mixture.