I know how to say 'I'm vegetarian' in several languages - Spanish, Italian, Russian, Turkish - and I've learned it and forgotten in some others. When I travel in China, however, I don't try to say anything; I use a little piece of paper with three Mandarin Chinese characters on it and that always does the trick. So, I'll share that to start with. Here it is - you can print it out and stick it in your notebook or wherever. I think it actually says something like 'I am a vegetable eating person', but it always works. You'll invariably be given vegan food too, not just lacto-veg.... and definitely no eggs.
Now, instead of repeating what's already out there on the web, I'm putting some links here to other sites which have the phrase 'I'm vegetarian' or something functionally similar, in various languages....
Catly at CookAdvice gives a variety of phrases in different languages with some phonetic/latin transcripts for languages with difficult scripts.
... and there's some good stuff at WikiHow - again with some transcription (and also some in Bengali script, which might be good for making cards).
For how to do it in Japanese, have a look at this page from the International Vegetarian Union, which has phonetic transcriptions and some script to copy and print.
There are heaps of phrases in different languages at Green Earth Travel, although I found some of them a little difficult to figure out. But have a look - the one you need might be there.
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