Hi Morghan!
You can absolutely get dinner reservations at Walt Disney World for your large group. I have been the planner for our family reunions for years and we had 16 for many of those trips. The important thing with large groups is to get reservations for all your dining as soon as you can once your 180 day mark hits.
The first thing you will want to do is to read through all the awesome
Dining Choices in all of the four parks and resorts. I suggest you look at getting one table service reservation per day but don't overload your days with a lot of heavy eating.
My next big suggestion is to call the dining number 407-939-3463 with your plan and let the cast member help you with the large group reservations. Sometimes on the website you can't find a time for large numbers. I found out that if a restaurant doesn't have a large enough table for your number they will not let you reserve. So many guests want to be at one table and they don't want to disappoint them. The cast member can work around this to get you a time for the entire group.
Which now brings me to my final thought. Do NOT worry about sitting at one big table. The reason is simple in that most places don't have them and your wait will be forever meaning you will miss park time. If you all were to sit at one large table you would not be able to talk with more then a few people anyway unless everyone yelled up and down which of course would not be enjoyable. My family had a tradition to accept any number of tables for our group. We would carry name cards and as soon as we knew the table configuration we would draw names for each table. It was fun and a great way to enjoy different family members at each meal. We included the children in this as well and they loved it!
Enjoy your upcoming Disney vacation. It is the perfect place for all generations to come together for a magical time.