"No, I am not Arab."
Translation:Nem, én nem vagyok arab.
Whether it's a positive or a negative sentence has no influence on whether you use vagyok or not. The trick is this: when you have a sentence saying "I am / you are [something]", then you'll have to express it with a form of van, except for the third persons.
Let's try it with my all-Arab family:
Arab vagyok. Te is arab vagy. Ő arab. Arabok vagyunk. Ti arabok vagytok, és ők is arabok.
Generally I would van rather translate with "there is" and vannak with "there are", since those words are not used for quality descriptions ("The car is red"), but for matters of placement and existence ("There is a car behind the house.")