Is it possible to become a wizard?
It is possible to study magical lore. It is possible to learn spells and incantations, and it is possible to learn the tricks that stage magicians and parlor magicians perform. And it is possible to call oneself "wizard."
It is not possible to perform actual magic. That is a fantasy.
On the anime Avatar it is possible to become a Fire Bender - which is virtually the same as a fire wizard. In real life: there is no evidence that magic of any type actually works, so while you can call yourself a "fire wizard" if you like... there's no State Board of Fire Wizardry to insist that you pass a practical examination first... no, it's not really possible.