I am thinking about getting into this game mainly to play solo, but I am tournaments. First, how popular is the game still? Do stores still even have tournaments? Also, how does a tournament for a co-op game exactly work?

There are no sanctioned tournaments. I hear a group in Europe does tournaments based on final score and some other criteria.

There are once-a-year official events made by FFG and hosted by local game shops. In those, FFG typically puts out a new scenario and you get some extra swag, usually an alt-art card and a playmat.

There is a competitive format which is based on 2 teams trying to complete the same scenario in the same amount of time, but it has virtually no following.

I would wager the game is as popular now as it has ever been.

There is also Keeping Count, the semi-competitive format that cardboard of the rings created:
https://cardboardoftherings.com/keepingcount/

The Race Against the Shadow tournament format is very enjoyable.

Rules:

https://images-cdn.fantasyflightgames.com/filer_public/67/af/67af4664-f4be-4a87-9054-76dfe63b850e/race-against-the-shadow-rules.pdf

We held a Race Against the Shadow event a while back at the FFG Event Center and I wrote an article about it:

http://www.cardgamedb.com/index.php/index.html/_/lord-of-the-rings/the-race-against-the-shadow-r974