We Brits can rejoice! Disney have a dedicated Shop Parks app, but it's only available in the USA. So we are unable to shop online (even while at Disney World) because the app is only available on the USA app stores.
Today, Disney updated the My Disney Experience app to include the Shop Disney Parks functionality. So now anyone with the My Disney Experience app can browse all the items that were for sale on the previous Shop Parks app.
Sadly, this does not mean you can buy things from home and have them shipped here. They will only deliver to the USA or to the parks for collection when you're there.
Login to your My Disney Experience app and scroll all the way down to the new shopping section.