Preheat oven to 450 degrees F (230 degrees C). Wrap beets tightly in aluminum foil.
Roast beets in the preheated oven until tender and easily pierced with a knife, about 1 hour.
Cut off the top and bottom 1/2-inch of the corn ear. Moisten with water and place on a microwave-safe dish. Cook in the microwave on high, 3 to 4 minutes. Let cool until easily handled, about 5 minutes. Slide husk off the cob and cut off the kernels with a sharp knife.
Combine corn, chickpeas, tomatoes, lettuce, edamame, celery, and carrot in a large bowl.
Whisk lime zest, lime juice, olive oil, green onion, garlic, cumin, chili powder, and salt in a small bowl until well-combined. Pour vinaigrette over chickpea mixture. Refrigerate salad until beets are ready.
Unwrap beets and cool until easily handled, about 10 minutes. Peel, dice, and stir into salad. Chill 1 hour to overnight to develop flavors.