Yes, the WDW resorts are well prepared to assist families with shipments.
If you will be staying at the Coronado Springs Resort, there is a business center that will help you mail your package home. For other moderate resorts, visit the Front Desk.
There is a handling charge based on the package's size for packing and labeling done by a Cast Member.
Disney Mom Diane explains her experience with mailing packages
here.
The Front Desk usually has boxes. If not, just ask the gift shop for an empty box. You can then package your items and put the FedEx or UPS label on the box. With airline baggage fees costing extra, shipping items home is a great way to save money.
Wishing your family a magical vacation!