Hey Judy!
I did this last year when I was staying at one of the Disney World Resorts that has a Business Center. Only Disney's Contemporary Resort, Coronado Springs Resort, Grand Floridian Resort & Spa, Yacht Club Resort, and the Boardwalk Inn have Business Centers.
The Business Center has Postal services including UPS which made it super easy to drop off and use my account. The center makes it easy, but it's certainly not the only way. Disney Cast Members can help you accomplish this with very little effort. I've never had a Cast Member even blink when I ask for shipping help.
I've also shipped boxes home from another Resort using the Front Desk. They took care of shipping it for me, but I did not ask if I could use my UPS account number when I did it that way.
It is much easier to ship items to the Resort and back home rather than lugging them around! Especially when you have little ones with you it will make for a much more relaxed parent while traveling! We've all seen the crazy lady, sweating & crazy eyed dragging heavy luggage through the airport while trying to keep track of several kids...yup, that was me once or twice! Nobody wants that!