Being a vegetarian or vegan in Japan can be tough. You can't just roll up to a place, say "I'm a vegetarian," and then get food free of beef, ham, bacon, shrimp, fish flakes, etc. You literally need to run down a list of things that can't be added to your meal. And that's because there's still limited understanding of what it means to be a vegetarian.
Over time, one gets used to it, but from time to time, you might forget to say "Katsuo mo tabenai." (I don't eat dried fish flakes either.) And then voila there's your beautiful yaki udon absolutely covered with katsuo.
Fortunately there are some useful sites and tools out there to help you find veggie restaurants, like Vege-Navi, Vege Tokyo, and Happy Cow. Other sites help you navigate your way through non-veggie restaurants. I posted this a few years back.
I just discovered a fantastic new post on the website Just Hungry. It includes a set of printable cards indicating specific dietary preferences and restrictions. The cards are bilingual (English and Japanese) and cover veggies, vegans, pescatarians, and others. Here's a sample for lacto-ovo vegetarians:
Click here to head to the post.
Keep in mind a sizeable number of restaurants will still not be able or willing to make adjustments for you. But plenty will. The key is being clear and polite up front, to minimize miscommunication.
I think in general, for restaurants and other businesses, the language barrier creates a lot of misconceptions. It's not like they don't want to help you - it's more a matter of their not wanting to misunderstand you or make a mistake. Plus there's a high inclination in general to avoid hassle and conflict.
So I think tools like these cards, which are written in crystal clear Japanese, could be very helpful. Itadakimasu (Bon appetit)! :)