So the question "When was Disney World built?" isn't simple: it was built over decades, starting with The Magic Kingdom which opened in 1971.
Long before Disney World was built, Walt Disney had grand ideas about an enterprise to follow his successful Disneyland built in Anaheim California. Disneyland had opened in July 1955; as early as 1959, Walt Disney began to look for land for a "Florida Project". Once the Orlando location was chosen -- inland, to limit the risk of hurricanes-- Disney acquired 27,400 acres (43 sq. miles), starting in 1963, using dummy corporations to keep the project quiet while the land acquisition went on. Walt Disney wanted the vast acreage not only to build on, but also to control against the sprawl of hotels that had sprung up around Disneyland. In 1965, after news leaks about the land buys, an announcement was made.
Walt Disney's vision included not just amusement parks but also a futuristic city called EPCOT, the Experimental Prototype Community of Tomorrow. However, plans for EPCOT were put on hold while the Magic Kingdom was built first.
Walt Disney didn't live to see the opening of even his first theme park in Florida: he died on December 15, 1966. His brother, Roy, took over the construction of the park.