I was like scratching my head too over this.
Click the Login link in the upper right of the site, it then loads up a Discord page, if you have a Discord account login, otherwise register and create one.
Once logged into Discord, which will now have you logged into the SG site, click the link below I've pasted here which will take you to the SG-DXXI House and you should see a section to fill out the application;
Hello I am 39 Retired Esports player 15 tournament wins from 2000 - 2005. I play more casually now, but still seeking high level game play with like minded people. I main Bangalore and have her 20 Kill , Triple Triple and Quick Executioner Badges. I like to shoot the * and have fun, I play Apex daily.