There is no application process for creating an Organization on Melee and begin running your own tournaments on Melee. You can create an Organization here. You will need to choose the name of your Organization as well as provide an email address to be used as the Organization’s contact email address for anything related to the Organization specifically. 

Once your new Organization is created, you’ll gain access to its Organization Dashboard as well as be able to create and run tournaments under the Organization. It’s important to read Melee’s Terms of Service and Entry Fee and Payment Policies before you start running tournaments.