flatlandcycle is correct - Reedy Creek Utilities Company (a wholly-owned subsidiary of Disney) operates several utilities serving the Reedy Creek Improvement District (which, surprise, happens to include Disney). Although this includes an electric system, RCUC only generates about 20-30% of the electricity used in the district. The remaining amount of electric power (70-80%) is purchased through numerous lease agreements with other power companies in Florida. If you want to know more than you ever wanted to about the electrical and other utility systems servicing WDW (which I did, because I'm a glutton for punishment), you can take a look at the annual report for utility systems on Reedy Creek Improvement District's website - the most recent one is for 2015.
So what does that mean? It means that a large percentage of WDW's power is coming from outside its own utility system and may be affected by disruptions that those service providers are having. I say may because I don't know whether, in a situation like this, RCUC can generate enough electricity on its own to cover basic functions (or more) without purchasing or relying on outside power. While it's all horribly complicated, the simple test is as follows: look up and see if the lights are on. If so, things are pretty good and I wouldn't worry too much.