Koryouri Kuhara (小料理 久原)
Koryouri Kuhara isn't a vegetarian restaurant, in fact, its specialty is fish and seafood. However, if you're a vegetarian and your friend is a fish-lover, or vice-versa, it's a great compromise as everyone can find what their heart's desires.
For pescatarians, you'll find fresh sashimi and grilled fish. If you don't fish either, worry not, you can have, for example, some seasonal vegetable tempura or stews, vegetable assortments, etc.
Adenia is a French bistro which specializes in steak so it's definitely not a vegetarian resto, but if your friend is a hardcore meat lover, you can both find something you like.
For lunch, if you're not a pescatarian, I would avoid going as there are only fish or steak/burger plates. If you do eat fish, the fish place is ￥1,200.
For dinner, unfortunately, there are no vegetarian main dishes (there is a daily fish plate though), but you can try a variety of starters and it'll make for a tasty and amazing meal. There is, for example, butternut squash honey nut toast starter or Greek-style marinated vegetables, etc.
This restaurant is a high-end Kappo restaurant which serves seasonal ingredients in a cozy and intimate setting. If you don't eat fish, you should probably skip this restaurant altogether.
The chef's omakase course is ¥22,000 and up. They serve a series of elegant small dishes and they are prepared right before your eyes.
When making a reservation, you should let the staff know you don't eat meat so that you don't have any unpleasant surprises.
Nepalico is a recently open modern yet cozy Nepalese restaurant you should definitely check out while in Shibuya! What is more, it's more than affordable and delicious.
Luckily, this place too has tasty vegetarian options, such as bamboo soup and curry rice.
WE ARE THE FARM
This place is a true treasure in the city. It serves fresh and tasty vegetables in a variety of ways, and the atmosphere is cozy and stylish. The portions are huge, so don't worry about leaving hungry after only having vegetables, but they do also serve meat. Main dishes are around ￥2,000-￥3,000.