Step 1: Set up your grill for indirect grilling and preheat to medium.
Step 2: Combine the green and purple cabbage and grated carrot in an aluminum foil drip pan and position in the center of the grill grate away from direct heat. Toss 1/2 cup of wood chips on each mound of coals. Cover the grill. Smoke the cabbage for 12 to 15 minutes, stirring once. Do not allow the vegetables to cook. Remove from the grill and let cool. Transfer to a large mixing bowl.
Step 3: In the meantime, in a small mixing bowl, combine the mayonnaise, vinegar, horseradish, sugar, lemon juice, salt, pepper, and celery seed and whisk until smooth. Pour over the cabbage mixture and stir gently to coat. Cover and chill if not serving immediately.