Everybody seems to agree that endocytosis is an energy-using process, and as such requires ATP hydrolysis. However, which particular step requires it? More precisely, which 'molecular machine' involved in endocytosis requires ATP? I cannot seem to find a good answer in the literature.
Endocytosis (specifically you asked about clathrin-mediated endocytosis) is indeed an energy consuming process. the coating of the vesicle may be "spontaneous", but the pinching off of the vesicle, the uncoating of the clathrin, and the transportaion of the vasicle inside the cell all require ATP/GTP hydrolysis.
Dynamin - dynamin forms a spiral around the neck of the vesicle, and uses GTP hydrolysis which in order tighten the coil around the vesicle neck causes it to break and results in the pinching off of the vesicle from the parent membrane.
The uncoating of the clathrin is also ATP/GTP consuming. proteins such as hsp70 chaperon and auxilin is thought to be involved in the ATP hydrolysis. the uncoating process starts after the pinching off of the vesicle.
the tranportation of the vesicle will also cost energy, proteins such as the Myosin family are considered to transport endocytic vesicles into the cell, and use ATP hydrolysis.
Dynamin - Wiki
Myosin - Wiki
Uncoating process - Molecular Biology Of The Cell ed.6 p.701