I work for a parking company that handles many Valet Parking locations. If a Valet Parkers damages your vehicle, they are liable under Bailment Law.
All legitimate companies (including Disney) carry Garagekeepers Liability Insurance, but with a deductible. This means that Disney is responsible to pay out directly any amount UNDER the deductible and the Insurance company OVER the deductible. I would assume that Disney has a very large deductible (ours is $25,000.00).
Again, there are many "fly by night" companies, but most legitimate companies will pay for the damage their valet cause. As said above, there will be times when there is doubt as to who is at fault. The best rule is inspect your vehicle when the valet returns to you. If there is any damage, have them complete a claim form and take pictures. If there is damage, DO NOT LEAVE without filing a claim - this will raise the issue of WHEN the vehicle was damaged. If there is a claim, ask to see the entire ticket, becasue most valet companies inspect the vehicle prior to parking it (the example above from the Swan/Dolphin) and then note any damage on the part of the ticket they keep. If there is no damage noted, this will prevent them from claiming the damage was pre-existing. If the damage was not noted on the ticket and you report it BEFORE you leave, there is documentation that the valets caused the damage.
I do not believe you will have an issue with a reputable company like Disney if they do damage your car. When I go to
DVC I always use valet parking and would not hesitate at all in the future.
Now with regards to Rental Car companies, when a vehicle is damaged they usually impose a "loss of use" fee and administrative charges. When we deal with the Rental Car companies after damaging one of their vehicles, we inform them that we do not pay these charges; we have not pay any to date.