It's a combination. Obviously Walt Disney Imagineering designs these projects. WDI is on site for the entire construction for overview, and to do many of the smaller details. The major construction is often done by the Disney owned Buena Vista Construction Company. You can often see their trucks complete with Mickey logo driving around property. Outside contractors are also used often, because the BVCC can only handle so much. For example, Pop Century was built by a combination of the Buena Vista Construction Company, Hardin construction and Centrex Rooney. Disney also operates two separate painting divisions, one being the Buena Vista, and the other being the Disney Powerhouse Painters.
If you see smaller details, such as murals, painted rockwork, and intricate details such as the fake mold on many of the buildings in Africa and Asia at AK those are almost always done by WDI.