I linked the english page but as you can see they aren't spending much on a modern and user-friendly UI. They seem to be in transition to a more modern one, but that appears to have zero english translation :p I'm not sure they even have any foreign language material, so you're probably not missing out on too much.
By card do you mean an actual physical item, or just access to their stuff?
The Hungarian National Library OSZK requires in-person registration. They even need an ID and take your photo. Though I'm not sure the pass even comes with online benefits. The Hungarian Electronic Library MEK offers their 25K collection without any need to log in afaik.