Finding a very large reservation may be challenging, and you may have better luck calling Disney Dining (407-WDW-DINE) than using Disney's website or app.
Via Disney's website or app, you can also try to book enough smaller reservations to meet your group's needs. If you do that and then ask upon check-in to be seated together, Disney will do what it can to accommodate.
Also note that buffets and all-you-care-to-eat venues tend to work better for large groups.
If you have a large group and would like to use different methods of payment (Disney Dining Plan for different rooms, cash, credit card, etc.) for different subsets of your group, that's fine. You can still make one very large reservation and have the server arrange the "split checks" appropriately when it is time to pay. It will be a bit of a pain, so be patient--but the Cast Members are used to it.
Finally, keep in mind that Disney automatically adds an 18% gratuity for any table of 6 or more people. Be sure to check your bill to make sure you are leaving the gratuity that you intend to provide.