How did they know it was Vito

Who murdered Fanucci. Nobody had seen who killed Fanucci (except a toddler Sonny who somehow managed to follow his father over the rooftops), and so any fool could claim that he was the one who killed him. Vito had never admitted to anyone that he killed Fanucci, nor did Clemenza or Tessio talked about it with anyone, yet the whole neighborhood knew Vito had killed him. How?

Word of mouth, the fact that Vito told Clemenza that he had lost the gun and paid him for it. Probably started with a rumor (albeit a true one) and just snowballed from there.

Women.

They talked among themselves. Maybe one of Fanucci's girls or his wife saw the murder but knew better than to say anything. But girls being girls, they talk and then it spreads through the streets.

Or maybe not.

Maybe Vito talked with Maranzalla out of respect and received his blessing to continue working the district in exchange for ridding the neighborhood of a rat and maybe a tribute to The Black Hand. And things just fall into place after that.

Keep in mind that the novel states that Clemenza and Tessio didn't speak with anyone about it. So the rumor could not have been spread by them.

I very much doubt it. If Fanucci's daughter had seen Vito killing her own father she would have likely said this to the police. At that point they did not need to fear Vito as he was still a nobody. It was only when the whole neighborhood knew he was the one who killed Fanucci that he was treated as a "man of respect", and this took a couple of weeks.

Logically, since Vito moved in immediately or almost immediately to take over Fanucci's rackets, and no one else challenged him, everyone would assume he'd killed Fanucci.

Most probably TB. Fanucci is murdered and then Vito, Clemenza, and Tessio start showing up in Fanucci's prior capacity. The people in the neighborhood put 2 and 2 together.

The novel says nothing about it. Only that after some time, some shop owners paid him voluntarily.

The novel said, "Although they [Clem and Tess] never spoke to anyone [about Vito killing Fanucci], the whole neighborhood, within a few weeks, also knew. Vito Corleone was treated as a "man of respect" by everone."

True TB. It also states that Tessio suggested to Vito and Clemenza that they could take over Fanucci's customers. In addition, it states that part of Vito's murder strategy was to have witnesses "to show that he [Fanucci] had left the Corleone home safely". That Fanucci ended up dead soon afetr and that Vito murdered him didn't require much math from the neighborhood's inhabitants.

Well, Clem and Tess knew that they only had to give Vito a fraction of what Fannucci wanted, and that he would take care of their problems with Fanucci. This was kept confidential among Vito, Clem and Tess (supposedly).

When Fanucci turned up dead, there were Vito Clem and Tess, with Vito having established himself as top dog among the three. Vito also suddenly had a wad of cash, a new rug in his living room, and was running the clothing etc. that Fanucci claimed was his. Also the neighborhood knew well it was rid of Fanucci. Certainly his death was public knowledge, and the cops themselved probably reported to the papers that Fanucci had been shot several times at close range.

I think it is very significant that suddenly Fanucci was out of the lives of the Abbandando family. As we know Genco ended up being Vito's consigliere, and it is not hard to imagine that Genco and his father (with whom Vito was close) were relieved to see Fanucci gone and Vito prospering. It probably was no leap of faith for the Abbandond0 family to credit Vito with getting rid of Fanucci, and it would be like Vito to neither confirm nor deny this to people coming up to him to give him credit. Also, Vito filled a vacuum left by Fanucci. Instead of terrorizing Italians he built a network of friends by doing them favors and then getting favors back in return. It is telling that the fruit vendor would not take his money, and it is even more telling that as the olive oil business was growing, Vito told that landlord who had refused to let the widow back into his apartment, to ask around the neighborhood about him. He did, and the result was he came back to Vito's office completely terrorized.

Given all this, the legend of Vito among his paisan probably outran the truth. I am certain the word was Vito got rid of Fanucci and took over. Moreover he was always a protector of the Italian neighborhoods ...even through his old age when he made the drug deal that it would not be sold in the old neighborhoods....soething even Michael tried to enforce against Joey Zasa generations later.

Then again, maybe Manolo spread the word.

In a deleted scene DT, you see Manolo in the street talking to a couple of paisani as Vito passes near by and Manolo points at Vito and then runs his finger across his throat. He was probably tellig them that it was Vito who got rid of Fanucci.

I think we are to assume it was spread slowly, gangster to gangster, then to people in area. The book and the movie leave it open. It's a good question.

My guess is that people just assume that Vito took out Don Fanucci and now all the Little Italy store owners have to pay their tribunes to him. No one else could step up to the role of the neighborhood extortionist at that time.