Preheat a grill to medium. Prepare the chicken: With your knife parallel to the cutting board, slice each chicken breast in half horizontally without cutting all the way through; open like a book. Cover with plastic wrap and pound until 1/2 inch thick.
Make the garlic-tomato oil: Mix the sun-dried tomato oil and garlic in a bowl; season with salt and pepper.
Season the pounded chicken on both sides with salt and pepper and brush with half of the garlic-tomato oil; top with the zucchini, chopped sun-dried tomatoes, mozzarella and basil, leaving a 1-inch border around the edges. Roll up the chicken toward the pointy end; tie with 3 pieces of kitchen twine. Brush with the remaining garlic-tomato oil.
Brush the grill grates with vegetable oil. Grill the chicken breasts, turning, until cooked through, 18 to 20 minutes. Transfer to a cutting board and let rest 5 minutes, then remove the twine and slice.