I would like to know if I have to start a new game if I purchase the season pass after I started playing the vanilla game, or does the content show up dynamically in-game/as a separate mode?

Thank you!

The White March is just a new area on the map you already have. So you should purchase it as soon as possible, since you can start playing White March content almost right away.

It offers new characters and other content like items, which you'll want as soon as possible. All of them can be used normally in the rest of the game.

The companions from White March react to all the stuff in the game as well, so it's not really intended to be played last or anything.

Thanks!
So that technically still means it shows up even if I already have a game going?

Yes - you don't need to start a new game - you'll get a message from your steward and then it opens up