Japan is not a really easy country for vegetarian people because fish, and specially meat, consumed in tiny amounts, but they are everywhere. I would say that sometimes meat is used more as a condiment than as an ingredient. Also, in my experience, most of Japanese don't consider that things like the bonito flakes they put in the stock are "actual meat", and when you say that you don't eat meat, they actually understand that you don't eat steaks or sashimi.
Said that, there's a nice vegetarian Ramen shop in Tokyo Station:
Where do you live? Here in Minneapolis, where ramen is becoming trendy, many places have vegan alternatives.
That said, getting a really good umami flavor and rich texture in a vegan broth is hard - not impossible, but much harder than with meat. And frankly, most vegetarians are lousy cooks who don't pay attention to good textures and flavors, and most meat eaters are lazy cooks who don't learn how to get richness without meat doing the heavy lifting for them.
For vegetarian/vegan, a kombu-based broth can get that richness, but it'll take a lot of patience to make (this is also true of bone-based meat broth).
Search for shojin ryori, it's the food made at Buddhist temples (or temples of related religions), and probably it isn't the cheapest food. There are some well evaluated shojin ryori books on Amazon. Some temples will do vegan food, others will have eggs and milk, but I think you'll rarely find any kind of meat in it.
On the other hand, I'd love if they didn't put fish stock in everything, including ramen, I really like ramen. I found a ramen house where I live (Sao Paulo, there's a huge Japanese/Asian neighbourhood in here, and it's a great place to find vegetarian ingredients such as mushrooms) which offers one dish without fish stock, but they closed and reopened, and now I'm not sure they still have it on the menu.
And what has been said about vegetarianism in Japanese culture is true. They probably think fishes come from trees. I actually had an uncle (I'm from a Japanese family) who said he was vegetarian but ate fish.
It's pretty rare, and most so in the more traditional places. Even if there's no meat in the ramen, the broth is generally pork (occasionally chicken) based.
Funny story time: in my late and unlamented career as a translator, I was attached to a pair of US artists visiting Gifu's international arts and crafts fair. Both were vegans and expressed a desire to have vegan ramen.
I explained this to the ramen shop proprietor. He came out with a bowl of ramen with chashu (pork) on top of it. When I pointed out that pork does in fact count as a meat product, he plucked it off with chopsticks. I then said "Excuse me, I don't mean to be a bother about this, but I can't help but notice that you poured the broth from a pan marked Pig. Is there, I don't know, pork in the broth?" "Oh you read Japanese!" "Yep, literacy is a handy quality in a translator." "Do your guests read Japanese?" "No." "Then as far as they know it's hearty vegetable broth now isn't it!"
If you look out a miso broth then you’re probably OK. My wife has fish and shellfish allergies, but a surprising number of places were OK with that. The scary one for her is my favourite: kitsune udon. What looks like a block of fried tofu traditionally has bonito flakes mixed in before cooking.
While this is the "wrong" season for it (unless you are in the southern hemisphere) - you might want to try to cook a "nabe"[1] at home.
It's a brilliant, simple dish -- and typical for Japanese cuisine it can be magnificent in its simplicity if you are careful when selecting what to put in.
I would personally prefer some beef, fish, shell fish ... but there's nothing really inherent in the dish that dictates one need it. You'll need to amass things that in aggregate makes for a good broth -- and vegetables certainly can do that.
[ed: Ah, I was a bit quick googling. The recipe is great for the first part of the nabe -- but it doesn't mention that one way to end the whole thing, is, when all the other ingredients have been eaten, and have left their taste in the broth: Add ramen - and eat with the broth.]
If you live in the westwood area around ucla in LA there's a place that serves vegan ramen. It uses a soy based broth and it's really good. I'm not even a vegetarian.
In Japan, they usually think fish in vegetarian food is normal.
So you need to take care of the ingredients even when they clam they are vegetarian restaurant.
In the US every ramen place I've ever been to has had a vegetarian alternative. If you simply don't like the taste of meat (as opposed to religious or philosophical purposes) you can ask for it without the meat - you are likely to get meat broth but that tastes more salty than meaty. Outside of the US it is probably rare.
