Yes and no!
G-Loot is an esports platform that tracks your in-game stats and events. If you want to play the Missions and Brawls, the app needs to be downloaded, and it needs to be running in the background while you're playing to be able to track your game.
You do not use the app to play in the Tournaments and Showdowns, however. For those, you will enter the server indicated on the G-Loot website after you have registered to a Tournament or Showdown. Note: You will need to play Missions and Brawls to earn tickets, which will qualify you for the Showdowns.