In a large mixing bowl, combine the teriyaki sauce and salmon fillets. Coat the salmon well and let it marinate for about 10 minutes.
In a medium saucepan, bring the water to a boil. Add the jasmine rice, reduce heat to low, cover, and simmer for 15 minutes or until the rice absorbs all the water. Remove from heat and let it sit, covered, for 5 minutes.
While the rice is cooking, heat the vegetable oil in a non-stick skillet over medium heat. Add the broccoli florets, snap peas, and sliced carrot. Sauté for 5-7 minutes until the vegetables are tender but still crisp. Remove them from the skillet and set aside.
In the same skillet, add the marinated salmon, skin side down. Cook for about 4-5 minutes on each side, or until the salmon is cooked through and flakes easily with a fork. If desired, baste the salmon with some of the remaining teriyaki sauce.
Fluff the cooked rice with a fork and divide it among four serving bowls. Top each bowl of rice with cooked salmon, sautéed vegetables, and garnish with chopped green onions and sesame seeds if using.
Serve immediately and enjoy your Teriyaki Salmon Rice Bowl!