Preheat an outdoor grill for medium-high heat and lightly oil the grate. Preheat the side burner of the grill for medium heat.
Place broccoli, carrots, mushrooms, and zucchini in a large bowl. Drizzle olive oil over and toss to coat. Place vegetables into a grill basket.
Combine soy sauce, sugar, vinegar, sesame seeds, cornstarch, garlic powder, and ginger paste in a medium saucepan and whisk until combined. Place saucepan on the side burner of the grill and cook until it starts to bubble, about 5 minutes; reduce heat to low and let simmer.
While sauce cooks, cook vegetables on the preheated grill for 15 minutes, flipping every 5 minutes. Pour teriyaki sauce over vegetables and mix well to combine. Continue to cook until tender, about 5 minutes more.