Getting New Members
The most important thing for a organization is, no matter how silly it sounds, to have members. Getting additional members can be done by two ways: telling the world that you want more members and getting them to apply to your organization, or inviting individual players.
People are able to apply to your organization by default. If you don't wish for people to be able to apply to your organization, you can close your application process by going to the Preferences screen (in the Organization game menu), and selecting Options. From here you can disable the application process. No new applications will then be able to come in. Already submitted applications will not disappear though.
Whenever a person submits an application, you will get information through game notification. You can then go to the Preferences screen, and click on the link saying how many pending applications there are. This screen will give you a list of all the applications that you have received, as well as their level, score, last official organization, current organization (if any), and their explanation on why they should be accepted. It is then up to you to either accept, reject or ignore the person. If you select to ignore the person, he or she will not be able to apply to your organization again. This will not expire until the season is over.
Inviting new players are done by going to the player's profile, and clicking on Invite. You will then be able to enter a message why he or she is wanted. The invitation cannot be removed, so be wary who you invite or not. The player in question will then get a game notification about this, and can either accept, reject or ignore by going to the Preferences screen and clicking on the link saying how many invitations there are. Ignore has the same effect as it has for organization applications.