In Taiwan, vegetarianism is inextricably linked to local Buddhist practices. Buddhist vegetarianism in Taiwan excludes all meat and seafood, but also onions, leeks, and garlic. Many vegetarian restaurants also don't serve alcohol or caffeinated drinks, but this is not consistent.
Some Buddhist restaurants do not serve egg, but it has become more common to find dishes with egg at vegetarian restaurants in Taiwan.
Because much of the population eats vegetarian some of the time, there are a huge number of vegetarian restaurants and food carts, so finding veg food is pretty easy, overall. Vegetarian food is only becoming more popular, and vegetarian establishments are incredibly easy to find.
Vegetarian establishments may use dairy in some dishes, though a restaurant that describes itself as "pure vegetarian" probably won't. There are also "vegetarian bakeries" in Taiwan where you may be able to find a few vegan baked goods, though most items still have dairy. Regular bakeries may have one or two vegan items. A baked good that is often vegan is brown sugar cake, as many bakeries make it without eggs or dairy.
There may also be some confusion if you ask if something has butter. People may respond with something along the lines of "we use vegetarian butter", which probably means margarine. There is a Chinese word for margarine too, but people don't always use it.
If you aren't able to speak Chinese, you may have a really hard time if you go to a non-vegetarian place. It will be confusing to them if you try to explain to them that you don't eat animal flesh, broth, or seafood, but that you *do* eat garlic and onions. If you just use the Chinese word for vegetarian, that will be understood, but then you'll get food without garlic or onions. If you go with a Chinese speaker to help, you'll probably have more success, but there may still be some confusion.
In Taiwan, vegetarian food is almost a separate type of cuisine, and tends to be relatively light and sweet, though there is still a huge variety, and you can even find vegetarian "Western" food (often Italian) at some restaurants. This also means that non-vegetarian restaurants may have almost no vegetarian food, and that they don't really get the idea of "vegetarian-izing" a non-vegetarian dish.
If you go to a Thai or Indian restaurant instead, you can probably get some more strongly spiced vegetarian food. Thai restaurants in Taiwan seem to get more foreign patrons, and so are often a bit more flexible in making non-veg items vegetarian. Indian restaurants seem to have fewer vegetarian options than those in the west, but can probably be counted on to have something.
If you don't speak or read Chinese, there are two common symbols that mark vegetarian food. One is the lotus flower, and the other is the swastika. The characters for vegetarian food (su4 shi2) look like this: 素食
If you're veg and don't speak Chinese, my best recommendation would be to only frequent vegetarian establishments. If you're vegan, you should be able to avoid dairy relatively easily, since it will usually be obvious (ice cream, milk-based drinks, cheese). Dishes with eggs are often obvious, but if you learn enough Chinese to ask if something has eggs, that would be useful. You'll probably be going to a lot of food carts and buffet/self-serve restaurants, since English menus are incredibly rare except at non-Chinese restaurants.