Well, the title about says it all.  I've very recently started playing the LOTR LCG and am curious if tournaments exist and how they are played/scored?  Is there an official FFG page somewhere that I have missed?

There are no tournaments as of yet, and it might never happen. It appears the co-op aspect of the game has made the tournament idea a non-starter. Most people I talk to don't seem to mind either. Most prefer the idea of special fellowship events with participation prizes such as alternate art cards, and maybe an extra  "you managed to beat the quest!" Prize. 

The are ways to score each game, and the most current version can be found in the FAQ on the support page.

The co-op system makes a truly competitive tournament extremely difficult to execute. Essentially, you would need a judge at each table watching every play of every group's game(s). 

Since all of the players are on the same team, and motivated to beat other teams, and the encounter deck  is not a sentient opponent, there is just too much potential for cheating, unless there is a 1:1 ratio of tables and tournament judges.