The program gets its "volunteer opportunities" through Hands On Network. You sign up (through the Disney site), choose an opportunity (a day of volunteering), do the volunteering, get verified and then you get your ticket.
If you want your org to be considered as one of the beneficiaries of all this volunteerism, go here.
The Hands On Network has what are called Action Centers. They "reach 83% of the population, facilitating more than 30,000 projects a month. Our network also includes more than 70,000 nonprofit, faith, education and community-based organizations. There truly is something for everyone who wishes to engage in service. If you don’t see something you like, don’t worry - your local HandsOn Action Center will help you design, develop and manage a project that really gets your blood pumping!"