You can use UPS or FedEx.
It's easy and you don't need a car. And you can collect your purchases from all over property and ship them together.
Bring an empty shipping box from home- flatten it and pack it in the bottom of your suitcase. Also bring a FedEx or UPS label and packing tape with you (although the front desk can often help with the packing tape- and even the labels, I like to know I have everything I will need with me).
Reassemble your box in your resort room and add your purchased items throughout your trip. When the box is full (or your trip has come to an end

) - just seal up the box. Bring it and the label down to the front desk and they will weigh it for you and let you know how much it will cost.
I have accounts with both FedEx and UPS, so I just put my account # on the label and they (UPS or FedEx) will bill my credit card on file. It does not cost anything to have an account with either shipping company and makes life easier very often!
Disney will simply add your box to the next pickup by whichever shipper you use.
Another point I like to note- if you are shipping TO your resort before you arrive- make sure to NOT use the US Post Office. USPS deliveries go to a clearing house location before being sent to the resort. Your delivery confirmation will say it has been delivered, but it's not actually at the resort. Always use FedEx or UPS and your package will be delivered directly to your resort. (I learned this the hard way one Christmas trip years ago)