There are two kinds of technology and design issues at hand here. A website designer deals with its appearance, graphics, and user interface. They typically have much less to do with the "back end" plumbing that wires you up to your ADR's, tickets, and resort reservations. That back end is where Disney is having 99% of the problems with this new system.
There is a massive amount of data being thrown back and forth on this site, on a 24x7 basis, and the systems/software necessary to coordinate the kind of systems Disney needs are miles beyond the scope of just a website designer.
The problems seen during this rollout are occurring primarily because there's simply no software development lab in the world capable of providing even a remotely realistic "real time" load of data, users, and transactions such that adequate testing can be performed before rollout. You can simulate certain levels of input, certain volumes of traffic, but at some point, those simulations are rather "sanitized" models of real world conditions. The only way to assess a system of this size, unfortunately, is to start rolling it out for real on real people with real reservations. And I have to believe some poor IT soul or souls is presently on their 9th gallon of Pepto dealing (on probably a 24x7 basis) with all the issues this system has had and will continue to have until they get the kinks worked out. Mind you, every second of downtime is also lost revenue for Disney, so they have every incentive to get this thing fixed, too.
As a dev myself, I wouldn't want to be within five miles of the team tasked with rolling this out. It would be a world-scope effort with every minor glitch and hiccup magnified 1,000 times over.