[Columbine]

'i'm allergic to wheat' the sentence you want is 'komugi ni arerugi ga arimasu' or in Japanese: 小麦にアレルギーがあります. Consider printing this out or taking a written copy with you. You can then as 'osusume ha nan desu ka?' おすすめはなんですか (what do you recommend?)　and the waiter/waitress/chef can usually oblige. If you want to specifically ask 'Is this wheat/gluten free?', you can use komugi/guruten no haite imasen ka? 小麦／グルテンの入っていませんか
The specific term for Coeliac is simply セリアック病 seriakku-byou.

You will probably be able to eat kudzu/arrowroot noodles or rice noodles, but you'll have to avoid udon definitely and probably ramen too. soba are made with buckwheat.

In Japan, the vinegar used in sushi will definitely be rice based, so you may be ok with it, but be careful with soy-sauce. I hear some has it in but the one type that definitely doesn't is 'Tamari'. You'll have to be careful of soups, marinades and sauces as many will contain soy sauce.