Hi, Ilia! This is one of the most popular restaurants at Walt Disney World, so I understand your frustration in not being able to book something. There is always a chance of just walking up and being seated, but my advice for families that wish to do this is to try to go as early as possible or towards the end of the meal - like a late breakfast or a late dinner. If you go early, they may be able to accommodate you, but if the whole evening looks full, they will be able to tell you right away so you can choose an alternate dining location. If you go later, you may run into a situation where someone with small children couldn't stay up that late for dinner or they ended up not wanting to leave the parks to head over for breakfast. It could happen!
I always tell people to keep calling, and be flexible. Try for breakfast or dinner for each day of your stay, and ask them to check multiple times. Keep calling - up to the day you want to dine there. You never know when a last minute cancellation will open up.
I hope you get lucky and find an open table! Don't give up - as they say, "Faith, Trust and Pixie Dust!"
Have a great vacation!