In fact you can bring it. They check your bag very formally.
There is a large room like a canteen having tables and seats but no vendor it's called school lunchrooms in the lower level floor. You can bring your own food and eat at there. I did it (bread & fruit). Museum's staff recommend me to did like that. However, you may go there a bit later from lunch time to avoid a lot of students.
There aren't any really any vegan options in the museum. My wife and i are vegan and were just there. They do have just mayo though lol. They had a veggie burger but we didn't check the ingredients. You can leave thr musuem to eat and come back. We did bring some fruits and bars with us though and had no problems
You have restaurants inside but can't bring food. Really.
No..you are not allowed to bring food to the museum..but you can buy food from restaurants inside..you can find vegetarian food
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