The problem is dashi
Almost every savoury Japanese dish is built on a stock made from dried bonito, or from small dried sardines. Vegetable dishes, plain noodle broth, simmered tofu, miso soup — most of them contain fish even when nothing about them looks like it does. If you avoid fish, this is the single fact that matters, and it is worth asking about explicitly rather than assuming.
Ask about the broth, not the dish
The useful question is “does this contain fish stock?” — in Japanese, dashi wa sakana desu ka. Kitchens answer it accurately, because it is a question they understand. Asking whether something is “vegetarian” often gets a yes when the dish contains dashi, since the word is understood to mean “no meat”.
What is genuinely safe
Shojin ryori, the temple cooking served near large Buddhist temples, uses kelp and shiitake stock by rule. Some ramen shops now make a vegan bowl and say so on the door. Indian and Nepalese restaurants are everywhere in Japan and are the reliable fallback in any Japanese city, Nagoya included. So are the vegetable-and-rice sets at conveyor-belt chains, where every ingredient is listed.
Halal in Nagoya
There is a working Muslim community in the city and a mosque in Nakamura ward, and a handful of certified restaurants operate near it and around Osu. Osu is also where most of the city’s non-Japanese kitchens are, which makes it the practical place to start. Outside those, assume that broth contains pork and that cooking sake is used in almost everything.
Allergies
Wheat is in soy sauce, so it is in nearly every seasoned dish. Egg appears in batters and in the finish on many rice bowls. Buckwheat and wheat noodles are boiled in the same water in most soba shops. Serious allergies are best handled by writing the allergen on a card in Japanese and showing it before you sit down, rather than at the point of ordering.
Phrases that carry the day
Niku nashi de dekimasu ka — can you make it without meat? Sakana no dashi wa haitte imasu ka — is there fish stock in it? Butaniku to arukōru wa taberaremasen — I cannot eat pork or alcohol. — arerugī ga arimasu — I have a — allergy. Write the key one on paper before you go out; a written card outranks pronunciation every time.
Staff here would far rather be asked twice than get it wrong once. The awkwardness you fear does not exist on their side of the counter.
What this site can and cannot tell you
The dish descriptions here explain what a dish normally contains. They are not a statement about any particular kitchen, and no automated data source is accurate enough to be trusted with an allergy. Confirm at the shop, every time.