Peanuts and peanuts oil are not commonly used in traditional Japanese dishes.
But sometimes they are found in snacks and sweets - mixed with small rice crackers, in chocolates, just for example.
I read a few webpages about peanuts allergy in Japan, many of them warn about currey rice, snack food, salad, sandwiches as "they sometimes use peanuts products to add flavour". And there are various kind of breads with peanuts filling (paste, cream, etc.)
Peanuts is called ニ痴ツーニ段ニ鍛ニ団 (piinattsu) or ニ痴ツーニ段ニ団 (piinatsu) in Japanese.
Most food should be OK but since you are strongly allergic, you might want to make sure, you can ask at restaurants or shops.
ナス窶樞?堙哉痴ツーニ段ニ団窶堙俄?ケツュ窶忸窶堙姑但ニ椎槌停?ケニ樽ツー窶堙ーナスツ昶?堙≫?堙??堋「窶堙懌?堋キツ。
窶堋ア窶堙個食窶堙冷?「ツィ窶堙俄?堙哉痴ツーニ段ニ鍛ニ団窶堙「ニ痴ツーニ段ニ鍛ニ団ニ棚ニ辰ニ停?ケ窶堋ェナスg窶堙ュ窶堙ェ窶堙??堋「窶堙懌?堋キ窶堋ゥツ?
窶堙坂?堋「 ツ ツ ツ ツ ツ ツ ツ ツ ツ ツ ツ 窶堋「窶堋「窶堋ヲツ ツ ツ ツ ツ ツ
watashi wa piinatsu ni kyoudo no arerugii wo motteimasu.
kono tabemono niwa piinatsu ya piinatsu oiru ga tsukawareteimasuka?
(I'm severely allergic against peanuts.
Are any peanuts or peanuts oil used in this food?)
You can print it out and show it to the people are you asking.
The last line is 窶堙坂?堋「 (hai/yes) and 窶堋「窶堋「窶堋ヲ (iie/no), so that people can indicate with their finger which is their answer.
There must be a webpage to help people with allergy travelling abroad, but I cannot remember which site it was.
I read about it at japan-guide.com.
japan-guide.com forum - Questions -
If you use "search questions" function with "allergy, allergic" as the keywords, various posts are shown related to allergy.
I hope other members will post additional information.
And enjoy your trip! I hope you'll have a great time.