In a small bowl, whisk together 1/3 cup low-sodium soy sauce, 1/3 cup rice vinegar, and 2 tablespoons of the sesame oil; set the sauce aside.
Heat a large skillet over medium-high heat and add the remaining 2 tablespoons sesame oil until shimmering.
Add the chicken pieces to the hot skillet and cook, stirring occasionally, until cooked through and browned, about 8–10 minutes.
Add the thinly sliced ginger and garlic to the skillet and cook for 1 minute more until fragrant.
Pour the reserved soy sauce and rice vinegar mixture into the skillet, then add the chopped zucchini and sliced bell pepper.
Stir and cook until the vegetables are tender and the sauce thickens slightly and coats the chicken, about 5–10 minutes.
Remove the skillet from heat and stir in the fresh basil and 1 teaspoon chili flakes; toss to combine.
Serve the three-cup chicken over rice.