Vito was made when he killed Fanucci. He avenged his father way later.
Made by whom? Did he make himself?
The thing is, he could only have been made by another Don in New York, and then would have subsequently become part of that family.
Exactly: made by whom?
After his conversation with Don Roberto, only his neighborhood recognizes him as a man of respect. Subsequently, Vito designated Clemenza and Tessio as capos, but only after his alliance with the Italian bootleggers which then led to a spurt of his organization's growth. Unfortunately, there is no mention in the novel of Don Ciccio's murder or Vito's return to Sicily. The only plausible explanation which the film provides is that once he murdered Don Ciccio that Don Tommassino inducted him (of course, that assumes that Tommassino was already Mafioso).