Hi David!
There are some great options at Downtown Disney to accommodate large parties for dinner.
The key is to reserve as far in advance as possible. I'd recommend that you call in directly and speak to a Disney Dining Cast Member for assistance. They will be able to assist you with confirming a single reservation for a party with more than 14 people. Simply call (407) WDW-DINE.
Amongst my top recommendations would be:
Paradiso 37, Taste of the Americas
This is hands down, my personal favourite location to dine at Downtown Disney. I've enjoyed meals with many large parties here so much that I've returned with my family to celebrate special occasions. The staff and service are stellar, the food is fantastic and the atmosphere lends well to a large group.
Bongos Cuban Cafe
This lively, authentically Cuban dining location is a wonderful restaurant throughout the week, and a happening hot spot on weekend evenings featuring live music.
Portobello
This casually elegant Italian restaurant serves up delectable, authentic dishes in a lovely atmosphere. With multiple, separated dining areas within the restaurant, a larger party can often be easily accommodated.
I've provided links to these locations for you, so that you can read more about the restaurants and peruse their menus.
Thanks so much for visiting the Moms Panel with your great question David!
Please feel free to write me back any time if I may be of further assistance to you in your planning. I hope to hear from you again soon.
Wishing you a simply Magical time at Walt Disney World!
Kirsten