Hi, Jill!
This is a tough one at French Quarter. Coronado Springs, also a Moderate Resort, has many spaces as they cater to the Convention Crowd. Not only is French Quarter NOT suited for meetings, it is the smallest of the Moderate Resorts (which is a good thing, until you want some private space!)
Ask at the Concierge Desk when you arrive. They might be used to this question and can help direct you to a quiet space. If they can't help you out, here are my thoughts...
-Give the Food Court a try! Avoid busy times like breakfast. I would guess that around 10:30 a.m. (once everyone is off to the parks) that it gets fairly quiet.
-Scat Cats lounge is just outside the Food Court. It is only open in the evenings, so perhaps you can gather there earlier in the day. I would definitely get the OK from someone first, so you do not get all set-up then be asked to relocated.
-If you can meet early morning, you might be able to gather some chairs under palm trees pool side. Most families won't begin swimming until later in the morning, so if you have some early risers in your group, this might work.
I'm sorry I cannot be of more help, but a meeting of 15 - 20 is a pretty good size group! You might also want to contact Disney Meetings. They specialize in organizing meetings for groups of every size. They can be contacted at
www.disneymeetings.com or by phone at 321-939-7129. I hope you find someplace to adequately accommodate everyone!