Whisk the lime, oil, garlic, salt and pepper until combined in a small bowl. Place the chicken in a 1-gallon plastic bag. Pour half of the lime mixture over the chicken; turn to coat. Let the chicken marinate in the fridge for at least 2 hours and up to 8 hours.
Arrange the scallions, onions, green peppers and red peppers on a large rimmed baking sheet. Pour the remaining lime mixture over the vegetables; turn to coat.
Preheat the grill to medium-high heat. Remove the chicken from the lime mixture and bring to room temperature. Sprinkle each side of the chicken with salt and pepper. Grill the chicken until cooked through, about 7 minutes each side. Grill the vegetables until tender, turning frequently, about 10 minutes for the scallions and peppers and 15 minutes for the onions.
Transfer the chicken to a work surface and cover with aluminum foil. Set aside the chicken for 10 minutes. Meanwhile, slice the onions and peppers and arrange them on a platter with the scallions. Slice the chicken crosswise into strips and transfer to the platter. Serve with the Cheddar and sour cream.