Happy Cow website for dining out in Tokyo, Osaka, or Kyoto https://www.happycow.net/asia/japan/tokyo/ If you were to add low carb, no gluten, and/or soy allergens to strict vegetarian, then things start getting trickier. AI can also help suggest vegetarian dishes available at national chains.

There are apartment style hotels in Tokyo, Osaka, and Kyoto with kitchenettes. It might be a good option for stricter dietary requirements.

Depending on how strict of vegetarian, big ryokans with buffet or kaiseki might not be great. Offhand, I'd guess the Hoshino chain which caters to foreigners to be a bit better about vegetarian dishes (along with more foreign friendly staff) and/or put allergen labels at their buffet dishes (but double check that). There's also some vegan/vegetarian friendly ryokan recommendations on reddit. Some ryokans do offer room only without meals/board if you'd prefer to eat out but always check for nearby restaurants open for dinner. Some onsen towns don't have many dining options open around dinner or breakfast time... or just not many non-ryokan dining options at all. Rakuten Stay also offers apartment/house style lodgings with kitchens/kitchenettes, some in onsen towns, so you can self cater https://stay.rakuten.co.jp/ They even have locations with private in-room onsens and saunas. ETA: They are unmanned/online check-in with no on-site front desk staff, so this isn't a good option for those who don't want to figure out Japanese (cooking) appliances, though they do have online instructions.

If your budget runs to the luxury range, it might be worth asking on the luxury ryokan thread Japan Luxury Ryokans - A Primer + Impressions Chefs at high end ryokans known for their food can usually accommodate dietary restrictions/preferences. However, IMO, few Japanese people would book a deluxe ryokan in a seaside area famous for their seafood or ranching area known for beef and ask for an entirely vegetarian meal for entire party so it might be useful to ask to narrow down your options.