What organisation is responsible for uluru?

It is not so much an "organisation" that is responsible for Uluru, as a group of people. The Anangu people are regarded as the traditional owners of Uluru, and the "ownership" of Uluru was returned to the Anangu in 1985. The National Park is now co-managed by a board made up of Anangu and Piranpa members, with their roles and responsibilities set out in both the Environment Protection and Biodiversity Conservation Act 1999 (EPBC Act) and the Aboriginal Land Rights Act (NT) 1976.