Pulse yogurt, basil, and onion in a food processor several times. Transfer yogurt mixture to a nonreactive bowl. Stir in curry powder and 1/4 teaspoon salt. Reserve 1/2 cup of the yogurt mixture for dipping.
Add chicken to bowl with remaining yogurt mixture; toss to coat. Cover, and refrigerate at least 45 minutes (or up to 6 hours). Meanwhile, soak twenty 8-inch skewers in water for 30 minutes.
Preheat grill to medium. Toss together bell pepper, zucchini, tomatoes, and oil in a medium bowl. Season with a pinch each of salt and pepper. Thread chicken and vegetables onto skewers, and transfer to a parchment-lined baking sheet.
Grill kabobs, turning occasionally, until chicken is cooked through and vegetables are charred, about 14 minutes. Serve with reserved yogurt mixture for dipping.