Back from Taiwan. I found it was extremely easy to eat vegan. If any of the restaurant staff speak even a little English, they will largely be familiar with the concept of the word vegetarian; veg is a popular monks' diet there. While this is Asia and therefore dairy is less likely to be part of the meal, it's better if you can explain to them that you don't want eggs or dairy. Also, if you specify vegetarian, you are intimating that you don't want garlic, onion, leek, chive, or scallion. These are considered lustful veggies and so the monks avoid them! In larger/buffet restaurants I just hunted down spicy pepper paste and put that on everything.
Even places which are called beef noodle shops may be able to accommodate you. One place we stopped for lunch gave me udon in water with vegetables. I added lots of soy sauce, vinegar, and hot pepper paste and had perfectly acceptable hot and sour udon. Also part of that meal and served to everyone: delicious tofu sausage and fresh bamboo shoots with garlic.
Shabu shabu/hot pot restaurants may be great places to try as well. Just make them give you broth instead of water for your individual pot. Go to the salad bar, choose base veggies for your stock, boil it, then add your choice of the many options (tofu, noodles, yuba, pumpkin, daikon, so many). Mix up your dipping sauce and enjoy. So delicious.
In Taipei, there was a vegetarian buffet restaurant that I really loved with absolutely tons of vegan options. Wonderful dumplings, turnip cake, fake meaty dishes (vegan orange chicken!), veggies, even vegan sushi and a number of make-to-order stations. It's in Zhongshan, called Ever Green, and you can see reviews on Happy Cow here
Things I loved:
Amazing oolong teas. Just at the south of Taipei, in Maokong, you can take an air gondola to tea houses for a lovely view and lovely tea and snacks.
Water spinach, yum.
Mushrooms. I had access to gorgeous mushrooms, steamed, stir-fried, and in clear soups.
Fresh warm soymilk, lightly sweetened, is a common breakfast. You can find it all over the place.
Proper fresh tofu.
Turnip/radish cake, so super good.
Seaweed potato chips!
Kanten (=vegan jello) with caramel sauce.
Fresh passionfruit and longan.
Desserts made of lotus root.