There’s a recent topic in German with I think exactly the same problem as yours: 🇩🇪 Ist der Fairphoneshop down?
For that person it seems the problem somehow just resolved itself after trying again a day later. Which isn’t much of a help in case the problem persists for you when trying again and again.
Some general things you might want to try out: use a different browser, delete cookies, use a private/incognito tab.
In addition, I think I remember once having seen a post with the suggestion to log into the shop account at the latest possible moment, but I can’t find that anymore. Or maybe it even was the other way around. Maybe it’s worth a try to put the item into your shopping basket before you logged in or afterwards (depending on which alternative you used to far).