Vegetarian food can be more challenging to find in Omoide Yokocho, but it is not impossible. Many izakayas offer vegetable-based dishes, and you may come across tofu or vegetable skewers at some yakitori stalls. For example, Horaiya has a special which is tomato and cheese skewer. Mayukichi offers kimpira gobo (burdock root cooked in sesame oil and soy sauce) and morokyu (cucumber served with moromi miso). Taro offers vegetable skewers, Komatsu has broiled veggies, Nogataya served potato salad, Fujinoya has grilled eggplant.
If you are a vegetarian, it is a good idea to hop a little to try a bit of everything in different restaurants. If you can't read or speak Japanese, learn Bejitarian de onegaishimasu and wait for the chef to offer you the veggie options.