Found it. If you are Admin you can add this custom field to the new ticket screen by going to Manage -> Forms. When you are viewing the list of forms you can click on Ticket Details, and then after Priority Level under the Form Fields section you can add a new field, call it "Organizations:", and set the type to choices. Click on config and it will let you enter a list of choices that can be selected when the ticket is created in order to specify what organization the ticket is for.