Whisk together the marinade: combine 1/3 cup olive oil, 3 tbsp lemon juice, 2 tbsp soy sauce, 2 tbsp balsamic vinegar, 2 tbsp brown sugar, 1 tbsp Dijon, 1 1/2 tsp onion powder, 1 1/2 tsp garlic powder, 1 1/2 tsp salt, 1 tsp each dried basil, oregano, thyme, 1 tsp paprika, 1/2 tsp black pepper, and 1/2 tsp red pepper flakes in a large bowl.
Reserve 2 tbsp of the marinade in a small bowl for basting and remove 3 tbsp into a separate bowl to toss with the vegetables.
Toss the peppers, zucchini, and onion with the 3 tbsp reserved marinade, then refrigerate until ready to cook.
Add the chicken to the remaining marinade, stir to coat, then refrigerate 4–8 hours.
If using wooden skewers, soak them in water while the chicken marinates.
Make the dip: whisk together 1 cup Greek yogurt or sour cream, 1/4 cup finely grated Parmesan, 1 tbsp lemon juice, 2 tbsp minced parsley, optional 1 tbsp chives, optional 1 tbsp dill, 1/4 tsp salt, 1/4 tsp ground cumin, and 1/8 tsp pepper; refrigerate until serving.
Thread the marinated chicken and vegetables snugly onto skewers, alternating pieces.
Grill method: preheat and clean grill to medium-high (about 425°F) and oil the grates. Place kabobs on the grill, cover, and cook 8–12 minutes, turning once halfway through, until chicken reaches 160°F. Baste once during cooking with the reserved marinade that did not touch raw chicken, then baste again after removing from the grill. Let rest 5 minutes before serving.
Oven method: preheat oven to 400°F and arrange skewers so they hang over a foil-lined pan. Bake 16–22 minutes, flipping halfway and basting once with reserved marinade, until chicken reaches 160°F. For more caramelization, broil a couple minutes per side while watching closely. Let rest 5 minutes before serving.
Serve the kabobs hot with the chilled herbed dip.