I downloaded Daggerfall from the site and started playing. A few days in, I got a note from this lady who wants me to meet her (I understand that this is for the main quest). She asked me to meet her in 'The King's Badger' in Longtale. However, such a location does not appear to exist in Longtale! Is this a bug, or am I overlooking something?

You're probably overlooking something

I talk to the NPCs there, none of them have that tavern on their list (they only have one, actually). I entered the only tavern I could find (and it's a small town), it's not the right place.

Re-read the note.

'For the purpose of our meeting, I will take a room at an inn, The King's Badger in Longtale of Daggerfall, for a month. After that, I will no longer be available. I will expect you as soon as possible.'

I asked the NPCs and searched the town. This inn is not present.