Hi Michael!
Traveling with a large group at Walt Disney World is so much fun! I coordinate a big reunion every year, so I am familiar with many restaurants who offer privacy, however, most are not located inside of a park. All of the in-park events I have booked have been in private event spaces. If this is something you may be interested in, Disney Event Services can help arrange something magical for your group.
Private rooms I have used in the past were
The Wave...of American Flavors or
California Grill at Disney's Contemporary Resort (both of these may have spending minimums),
Trattoria al Forno at Disney's BoardWalk. Many of the restaurants at Disney Springs have private or semi-private dining areas. Our group had an amazing dinner on the patio at
Chef Art Smith's Homecomin'.
Paradiso 37 also has a great outdoor patio space.
Maria & Enzo's Ristorante has a private room as well. More information on booking would need to be done by calling and speaking to a manager at any of these locations.
I hope I have at least sent you in the right direction to help make your group's next vacation a fantastic one.
Have a magical day!
Leslie