Combine the marinade: in a food processor, add olive oil, lemon juice, vinegar, cayenne, sugar, salt, pepper, garlic, parsley, and dill; pulse 15–30 seconds until well blended.
Divide the marinade: pour half into one large zip-top bag and half into a second bag.
Add vegetables to one bag and toss to coat thoroughly with the marinade; seal and refrigerate.
Add the diced chicken to the other bag, toss to coat, seal, and refrigerate; marinate both for about 4 hours, gently turning the bags occasionally.
While the chicken and vegetables marinate, soak the wooden skewers in water for at least 20–30 minutes to reduce burning on the grill.
Preheat the grill to medium (about 400°F).
Thread the marinated chicken and vegetables onto the soaked skewers, alternating pieces for even cooking.
Grill the skewers over medium heat with the lid closed, turning every 2 minutes, until the chicken is fully cooked, about 15 minutes total.
Remove from the grill and serve.