Cut zucchini in half lengthwise and cut crosswise into sixteen 1-inch pieces.
Cut the red onion into quarters and separate layers.
In a resealable plastic bag, combine chicken, zucchini, onion, oil, oregano, and 2 tablespoons vinegar; season with 1/2 teaspoon salt and teaspoon pepper. Marinate at room temperature for 30 minutes (or refrigerate up to overnight).
Heat grill or grill pan to medium. Soak eight 6-inch wooden skewers in water for 15 minutes.
Onto each skewer, alternately thread 3 pieces of chicken with 2 of zucchini and 2 of onion (see below). Grill skewers, turning occasionally, until chicken is cooked through, 12 to 14 minutes.
Meanwhile, make dipping sauce: In a food processor, blend feta, yogurt, mint, and remaining tablespoon vinegar until smooth. Serve kebabs with dipping sauce.