Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Place corn, onion, red bell pepper, and jalapeno pepper on a baking sheet. Drizzle vegetables with 2 tablespoons olive oil; season with salt and pepper. Rub to evenly coat.
Transfer vegetables to grill and cook, turning several times, until vegetables are tender and dark marks appear, 10 to 15 minutes. Set aside to cool.
Bring a pot of water to a boil; cook edamame in the boiling water until tender, 10 to 15 minutes. Drain and cool.
Whisk 1 tablespoon olive oil and vinegar together in a bowl.
Dice red bell pepper, jalapeno pepper, and onion. Cut corn off the cob. Combine vegetables in a serving dish; add edamame. Toss vegetables and edamame with oil-vinegar mixture. Season with salt, pepper, and basil.