Clathrin-mediated endocytosis is driven by a complex machinery of proteins, which assemble in a regular order at the plasma membrane. The assembly of the endocytic machinery is conventionally thought to be a continuous process of mechanistically dependent steps, starting from a defined initiation step. Indeed, several initiation mechanisms involving single proteins have been proposed in mammalian cells. Here, we demonstrate that the initiation mechanism of endocytosis is highly flexible. We disrupted the long early phase of endocytosis in yeast by deleting seven genes encoding early endocytic proteins. Surprisingly, membrane uptake and vesicle budding dynamics were largely normal in these mutant cells. Regulated cargo recruitment was, however, defective. In addition, different early endocytic proteins were able to initiate vesicle budding when anchored to a plasma membrane domain where endocytosis does not normally take place. Our results suggest that the cargo-recruiting early phase is not mechanistically required for vesicle budding, but early-arriving proteins can recruit the budding machinery into position at the plasma membrane. Separable early and late phases allow for a robust process of vesicle budding to follow from variable initiation mechanisms. Such a modular design could easily adapt and evolve to respond to different cellular requirements.