Bring a medium saucepan of water to a boil. Add 1 teaspoon salt to the boiling water. Add gemelli, and cook until al dente, 8 to 10 minutes. Drain, and rinse under cold water; set aside.
Heat 2 tablespoons olive oil in a large skillet over medium heat. Add onion and garlic, and cook until they begin to soften, about 2 minutes. Add carrot, and cook until it softens, about 4 minutes. Add zucchini, green beans, and corn. Cook until vegetables are tender, stirring occasionally, 10 to 15 minutes more. Season with salt and pepper. Transfer to a medium bowl to cool. Stir in remaining 2 tablespoons olive oil, vinegar, beans, tomatoes, and reserved pasta. Season to taste with more salt and pepper, if desired. Serve chilled or at room temperature.