Heat olive oil in a large pot over medium heat. Cook and stir celery, carrots, onion, and garlic in the hot oil until the onion is translucent, 7 to 10 minutes.
Pour vegetable stock over the vegetable mixture; season with Italian seasoning, kosher salt, and black pepper. Stir tomatoes, potatoes, and kidney beans into the stock mixture; bring to a simmer and cook until the potatoes are tender, about 40 minutes. Add kale and continue cooking until the kale wilts, about 5 minutes.
