Place the crabmeat in a fine-mesh sieve and press out any liquid. Remove any cartilage and set the crabmeat aside.
Cut the top quarter off of each tomato. Using a small spoon, scoop out the seeds and pulp. Drain the tomatoes cut side down on a plate lined with paper towels.
Meanwhile, in a large bowl whisk together the mayonnaise, sour cream, lemon zest and juice, tarragon, salt, and black pepper. Add the chickpeas, bell pepper, celery, parsley, chives, and carrot. Add the crabmeat and stir to combine. Divide the crab mixture among the tomatoes.