Combine vinegar, sugar, and 1/4 teaspoon salt until sugar dissolves. Trim ends of zucchini and squash; cut into thin ribbons with harp-shaped peeler. Add to vinegar mixture. Cover and chill 2 hours or overnight.
Bring a small pan of water to a boil; add garlic. Remove with a slotted spoon after 1 minute. Rinse under cold water; set aside. Add basil to boiling water; immediately remove and rinse under cold water. Reserve 1 tablespoon of cooking liquid. Transfer garlic and basil to a food processor, and add lemon juice, olive oil, reserved water, and the remaining 1/4 teaspoon salt. Process until smooth.
Drain squash, and divide among 4 plates. Top with cubed mozzarella, and drizzle with basil oil.