What do you need to do to get all of the League Card stars?

You get 5 stars for beating the main game, then you get a 6th star for beating Isle of Armor, and you get your 7th star for beating the Galarian Star Tournament.

You gradually earn more stars throughout the story of the game, and by the time you beat Leon and become the champion, you will have five. You earn one more by completing the story in the isle of armor by obtaining an Urshifu that can Gigantamax. And you earn one last one for completing the Galarian star tournament for the first time with whatever partner you chose. I highly recommend Hop for the first time. I know all of this information for a fact because I have all seven stars. You also earn one icon on your League card for each Pokédex you complete. It looks like a Rotom phone with a crown on it.