1) In the short term it would have been extremely risky and difficult for Barzini and Sollozzo/Tattaglia to run their business without Corleone assistance. As Barzini pointed out "[Vito] controlled all the judges". Without some sort of legal help/protection, as Sollozzo mentioned in the book whistle shhh , facing a 30 year sentence even a Sicilian might lose his nerve and start talking his brains out.

2) Barzini and Sollozzo weren't patient men. Barzini and Tattaglia had presumably taken the unorganized drug trade as far as they could take it without Vito's backing. They didn't want to wait another five or ten years to build up the capital and connections or wait for mores to change.

3) Sollozzo in particular felt personally humiliated. As he mentioned he did not intend to submit to another man's will. If Vito wouldn't assist, get rid of him and get somebody in there who would.

"When the snows fall and the white winds blow, the lone wolf dies but the pack survives."
Winter is Coming

Now this is the Law of the Jungle—as old and as true as the sky; And the wolf that shall keep it may prosper, but the wolf that shall break it must die.
As the creeper that girdles the tree-trunk, the Law runneth forward and back; For the strength of the Pack is the Wolf, and the strength of the Wolf is the Pack.