Well, you can doubt the existence of virtually every historical figure, especially when it is as remote in history. There is a French scholar who wrote a book in the 1820s to prove (jokingly, of course) that Napoleon was an allegorical figure who never existed. However, names rarely appear out of thin air, except when they have an obvious purpose. It is hard to see why the inventor of Zoroaster should have called him Zarathuštra, i.e. “he whose camels are old.” Not that this name is especially shocking in itself; giving deprecatory names is actually a common practice in various cultures, the underlying idea being to “turn away” bad luck (you can call that apotropaic practices, if you want to show off). But it is probably that if you invented a prophet, you would choose a more proper name (like, I don't know, “envoy/priest of Ahura”). Camels may have been an important economic reality, but it did not really translate in terms of symbolic reality into the Gathas (the oldest part of the Avestic corpus). Then, of course, the real importance of Zarathuštra, and the accuracy of the few and obscure written traditions about him can (and should!) be questioned.