Soak wooden skewers in water 30 minutes; drain. Meanwhile, bring 2 cups water to a boil in a small saucepan over high. Add fresh basil, and cook, stirring constantly, until leaves wilt. Drain and pat leaves dry with a paper towel. Place basil, vinegar, 1/2 cup of the olive oil, 1 teaspoon of the salt, and 1/2 teaspoon of the black pepper in a blender. Process until smooth, 2 to 3 minutes.
Preheat grill to medium (350°F to 400°F). Thread 6 chicken cubes, 2 red bell pepper pieces, 2 yellow bell pepper pieces, 3 onion pieces, and 2 zucchini pieces alternately onto each of 6 skewers. Repeat procedure with remaining 6 skewers, threading each with 6 shrimp, 2 red bell pepper pieces, 2 yellow bell pepper pieces, 3 onion pieces, and 2 zucchini pieces. Drizzle skewers evenly with remaining 1/4 cup olive oil, and sprinkle evenly with remaining 1 teaspoon salt and 1/2 teaspoon black pepper.
Grill, uncovered, until chicken and shrimp are done and vegetables are lightly charred, 3 to 5 minutes per side. (Shrimp kebabs will cook more quickly than chicken kebabs.) Transfer kebabs to a serving platter as they finish cooking. Brush evenly with half of the basil oil. Serve with remaining basil oil for dipping or drizzling.