Heat oil in a large pot over medium-high heat. Add potatoes, celery, carrots, and onion. Saute until slightly soft, 2 to 3 minutes. Add mushrooms and cook, stirring frequently, until onion is translucent, about 3 minutes more. Add chard and garlic; sprinkle with salt and pepper. Cook until chard is wilted, about 2 minutes. Add water to cover, bay leaves, and more salt to taste. Bring to a boil. Reduce heat and simmer for 35 minutes.
Bring a large pot of lightly salted water to a boil; cook fusilli at a boil until tender yet firm to the bite, about 8 minutes; drain.
Stir tofu, tamari, and bouillon into the soup. Simmer until bouillon is dissolved, about 10 minutes. Add a portion of cooked pasta to each bowl and ladle soup over before serving.