Probably the assumption that Fanuci was working alone is right.I will try to find any other hints regarding this in the book.
Turnbull, I think that both Genco and Clemenza opposite Vito's decision to go against Fanuci. So,they should belive that some bigger organization was behind him.Vito was the one that insisted on going against him.
Fanucci did have "friends". He also had some connections with the police. But Vito had thought everything through and decided that because there were some people in the area who paid no extortion money to Fanucci and that Fanucci had allowed his vengeance to be bought off in other cases, that therefore Fanucci was not really a true Mafiosi chief. Also Vito reasoned that Fanucci's friends were strictly hired per job and had no personal loyalty to him.
This was all pragmatic reasoning after the fact. Vito simply did not see a good reason to pay Fanucci money and felt the world would be better off without him. He was angrier about the attempt to extort money from him than he was fearful of Fanucci. This boldness and leadership is what bound Tessio and Clemenza to him for life.