If I make a member an admin of an organization will this allow that member to invite other members as well? Is this possible?
Anyone who is an admin of an organisation can invite other people so long as they're still an admin.
You can create a new organization through your accounts page, with the "Start an Organization" button at the bottom.
To add members to an organization, first go to the organization profile. You can get to the organization profile from the link in your board page or the link next to the title of a board. Then click "Members" in the sidebar. In the input field, enter an email address to invite or search for a current member. NOTE: You must be an admin of an organization to add members.
So after you've created the organisation, you can create your own Avon club of admins inviting new users to the group on the greater organisation's behalf.