Peel and devein shrimp, leaving tails intact; place shrimp in a shallow dish. Combine 2/3 cup olive oil and next 3 ingredients; pour over shrimp. Cover; marinate in refrigerator 4 hours.
Cut peppers in half lengthwise; remove and discard seeds and membrane. Place peppers, skin side up, on a baking sheet; flatten with palm of hand. Broil 4 inches from heat (with electric oven door partially opened) 10 minutes or until charred. Place in ice water 5 minutes. Remove from water; peel and discard skins. Combine roasted pepper, shallots, and next 3 ingredients in container of an electric blender. Cover and process on high speed 1 minute or until smooth. With machine running, add 1/3 cup olive oil in a slow steady stream, processing until blended. Transfer to a bowl; stir in mayonnaise, pickle, and capers. Cover and chill.
Soak 24 (6-inch) bamboo skewers in water 30 minutes. Remove shrimp from marinade; discard marinade. Thread 2 shrimp, 2 bay leaf halves, and 1 lemon wedge alternately onto each skewer. Place kabobs on rack; grill, uncovered, over medium-hot coals (350° to 400°), 4 to 5 minutes on each side or until shrimp turn pink. Discard bay leaves. Serve shrimp with tartar sauce.