I recently stayed at Shades of Green and did something similar.
I sent myself a USPS "Flat Rate" Priority Mail box full of stuff (and expected to use it for souvies, etc. on the trip, home).
Do I bring my own box to send it home in?
As in my case, my original box had no visible damage, so I peeled-off the old address label and stamp and affixed new ones for the return mail home (I did have a spare flat/untaped/unused box in my suitcase, in the event I needed it).
Do they ship via US Mail or UPS?
I used U.S. Mail (USPS) because of their "Flat Rate Box" option (it costs $8.95 or $12.95 depending upon which size box you use, regardless of weight/U.S. destination).
I would imagine if you used any other method, you would at least need a small postal scale to get an accurate weight -- plus, there would be no guarantees as to when FedEx or UPS would make their next stop at your resort's front desk (if, at all). Shipping via USPS is pretty "universal" at all WDW resorts as some CM at your resort makes a daily mail run to the Lake Buena Vista post office.
Do you have to pay by cash?
I purchased an $8.95 stamp (for the return package) from my local post office and enclosed it in my box I initially sent to WDW.
More information can be found at:
http://www.usps.com/shipping/flatrate.htm?from=priority&page=flatrate
And if you're wondering, everything worked like clockwork!