Drain tuna and add to medium bowl with 1/4 cup yogurt and 1 tablespoon mayo; mix well. Spread tuna mixture onto 4 slices of bread and top with remaining 4 slices. Cut in half.
For the salad: Cut apples into matchstick pieces and toss in a large bowl with rice vinegar. Julienne carrots and add to apples along with raisins, honey, 1/4 cup mayo, 3 tablespoons yogurt, celery seed, and salt. Toss to coat evenly.