In the following sentence, what will be the right preposition? He is ambitious ____ fame.
There are multiple possibilities depending on what you mean. That said, consider for.
Wiktionary lists this as definition 10:
In order to obtain or acquire.
Compare: He is hungry for fame. He strives for fame.
In this sense, fame is what he hopes to acquire. Why does he hope to acquire it? Because he is ambitious. So
He is ambitious for fame.