Originally Posted by LapLap

No soy, gluten or dairy foods? Is that right? (Sorry, no sugar either)

There are Starbucks everywhere in Tokyo, they have non soy dairy alternatives. You can take a little almond milk with you in your hold baggage until you get to go shopping, it’s fine to bring.

In your shoes, I’d consider making contact with a good will guide association and asking if they have a volunteer willing to go shopping with you to point out and help you identify products you can use. A gluten free section in a store won’t help you avoid soy and dairy. I think you need some proper guidance.
Another option is to visit the information desks of the large department stores and ask for an English speaker who can help you find these products in the “depachika” (food basements). If not immediately, if not in the first store, you WILL eventually find someone who can help you. All you need is a little patience, grace and persistence. Everything you want does exist... somewhere.

You are coming at a time when there’s a general push in Tokyo to better understand and cater to visitors’ needs, there’s never been a better time to go there with these particular challenges.