Update: Nintendo of America is aware of the issue and has published this tweet:
Reports have been coming in all day from both current and new Nintendo Switch owners who are struggling to access the Nintendo eShop to download their games, updates, and spend their generous Christmas gift cards. A number of users on social media have reported receiving the standard error message when they try to access Nintendo’s online store, which is presumably down because the Kyoto-based company’s servers are being hammered due to it being Christmas Day. Nintendo of Europe and Nintendo of America’s Twitter accounts haven’t addressed or acknowledged the issue on social media, which has riled some fans. No doubt Nintendo engineers are working hard to address the overload, but it would be handy for the company to acknowledge the issue and to provide a prospective timeframe when they believe that the Nintendo Switch Shop will be open for all to use.