The value resorts DO offer an allergy menu. However, they do not "guarantee" any item is completely free of the allergen.
Here is the statement from World Premier food court (the same statement is on the allergy menus at Music, Sports, POP and AoA):
About our allergy-friendly menu items: Guests may consult with a chef or special diets trained Cast Member before placing an order. We use reasonable efforts in our sourcing, preparation and handling procedures to avoid the introduction of the named allergens into allergy-friendly menu choices. While we take steps to prevent cross-contact, we do not have separate allergy-friendly kitchens and are unable to guarantee that a menu item is completely free of allergens. Allergy-friendly offerings are reliant on supplier ingredient labels. We cannot guarantee the accuracy of the contents of each food item. Allergen advisory statements (e.g., "may contain") are not regulated and therefore not taken into consideration when developing allergy-friendly meals. It is ultimately our Guests' discretion to make an informed choice based upon their individual dietary needs.
I spot-checked the menus at several resorts, both quick service and table service. The above statement seems standard across the board, regardless of resort category or restaurant type. My family has traveled for two decades with dietary needs, and I don't recall any "guarantees" on the allergy offerings. It's possible in 2015 the question was posed differently and/or the server answered differently, but I don't think the kitchen/prep situation has changed.