To prepare sauce, combine the first 6 ingredients in a small saucepan over medium heat; cook until the sugar dissolves and sauce slightly thickens (about 5 minutes), stirring frequently. Remove from heat; cover and chill.
To prepare wraps, steam cabbage, covered, 2 minutes or until crisp-tender. Drain and rinse with cold water. Drain and pat dry; set aside.
Heat canola oil in a large nonstick skillet over medium-high heat. Add carrot and next 4 ingredients (through 2 garlic cloves) to pan; sauté 5 minutes or until tender. Add cilantro, basil, and juice; toss gently to combine. Place about 3 tablespoons vegetable mixture and 2 tablespoons bean threads on each cabbage leaf. Top each with 3 shrimp; roll up. Serve with sauce.