Tempura is deep-fried battered seafood and vegetables. Most tempura places will have yasai tempura (vegetables only) on the menu. Eggs are used in the batter so it's not an option for vegans. The vegetables are also of course fried in the same oil as seafood. Skip the miso soup as it probably contains fish stock. The tempura sauce might too, so ask for it "shio de" (with salt) and use salt or soy sauce instead. Don't put the sauce or salt on the tempura, but dip the tempura in it. Common vegetables in tempura are nasu (eggplant/aubergine), Shiitake or other mushrooms, imo (sweet potato), kabotcha (pumpkin), renkon (lotus root), and onion. You'd be better to order a la carte because the sets usually contain miso soup and a kind of egg pudding made with fish stock. Tenya is a cheap fast-food-style tempura chain easily recognised by its blue and yellow signs.

Take a look at some recommended tempura restaurants in Tokyo.

Okonomiyaki is like a big fat pancake with cabbage and seafood or meat. Most places should be able to make it sans meat and seafood. Egg is used in the batter so it's not suitable for vegans. Usually, you cook it yourself on a hot-plate or it is cooked in front of you. For vegetarians, it's basically just cabbage and batter, but depending on the style of the restaurant, you can ask for other ingredients like soba noodles, kimchee, mushrooms, etc. One problem is that apparently some okonomiyaki restaurants use lard on the hotplate so try to find one which doesn't or be so bold as to ask them to use vegetable oil instead. Another problem is that okonomiyaki sauce sometimes contains fish, so you could ask for it without sauce. Also, they sprinkle fish flakes on it, so remember the magic phrase, 'katsuo-bushi nashi!' ("without fish flakes") Okonomiyaki restaurants also have other side dishes like enoki mushrooms cooked in butter. Monja, the Tokyo equivalent of okonomiyaki is similar, but sticky and runny.

Take a look at some recommended Japanese restaurants in Tokyo.