While we don't have any food allergy issues in our family, Vanessa, I have had a number of opportunities to speak with members of the Walt Disney Parks & Resorts Food and Beverage team at length. I can tell you that great care is taken to assure that guests are safe and cared for at all times. I encourage you to read this informative section of the website as it is a very thorough explanation of Disney's approach to serving guests with dietary concerns. Also, always note your specific allergies to the Cast Member where you've chosen to dine.
As for eating organic, unfortunately that is not as easily accomplished at Walt Disney World. My family, too, eats organic--about 75% of all of our groceries fall into this category. I can tell you that
The Wave has some organic options. Additionally, some restaurants serve produce grown inside
The Land Pavilion (though it's tough to say exactly where you can find it!). While I do not believe such produce is certified organic, I can tell you that instead of pesticides, they rely on ladybugs to keep insect predators at bay. I wish I could be more helpful on the organic front, but it's just not widely available at Disney as of now.