Finding good vegetarian food in Japan, China, Korea and Taiwan can be challenging. Most people do eat meat and fish and most dishes may include those things.
They may not always be noticeable either as they may have been prepared with say a fish sauce or perhaps a pork bone broth. That being said there are dishes that can be found that are suitable for vegetarians and vegans.
Vegetarian or not, food in general can be quite a challenge for some abroad. Here's a video where I go into the details about being a vegetarian in Taiwan, China, Korea and Japan.
How to say "vegetarian" in Mandarin Chinese
This will work in China and Taiwan.
You can always cook your own food, but the best place for vegetarians or vegans is Taiwan. Here you will find a large number of restaurants that cater to Buddhists who are vegetarians. They don't serve meat here and they don't cook with garlic or onions.
I think the food is pretty good and can range from buffet style and pay by weight, which is usually pretty cheap, to paying by the dish. I payed on average anywhere from about $60-130NT for the buffet style restaurants which is about 2-5 dollars.
In Korea you can eat dishes like Bibimbap, Bokumbap, Gimbap (usually has ham) and Suje bee. I ate a lot of Bibimbap. Sometimes they can put beef or ham in things, but you can request it without. I don't think Buddhists here are strict vegetarians.
In China you can find some restaurants like in Taiwan that cater to vegetarians, but not as many. I haven't lived in Japan yet and I have only visited, so I am not very familiar with it.
But I have heard from other vegetarians there that Indian restaurants are good places to find vegetarian food. I personally found some cheap noodle stands there that were meat free, as far as I knew.
Of course in all of these places you can find western style restaurants that may have vegetarian options. However, these restaurants can be pricey. Cooking your own food is often best.
Check the video out for more on vegetarian living in Asia.