You have several options. Anywhere you can purchase something at a park, they will gladly ship it directly to your house for you right then and there! It's very easy - they just give you a slip to fill out, and they treat it with great care. I have done that often. I usually do that when purchasing anything fragile - like glass or ceramic statues.
If when you get ready to pack up and go home, you find there are things you would rather not carry with you, as Judy said, you may bring all the things you want shipped along with the receipt to the resort gift shop, and they will package it all up into a nice box and ship that to your house.
Shipping is quick, and I have never had any trouble with it. Disney insures all items for the value and if anything is damaged, they will replace it for you. I had that happen ONCE and Disney was very fast, friendly and efficient at replacing the damaged items.
If you need to ship something that you did not get from Disney, there are a few shipping centers in the conference centers (the one I know best is in the contemporary conference center). They will package up your non-Disney items or items you wish to insure separately and ship those with any carrier.