Yes they do, and overnight as well.
The only pet car facilities on Disney property are at The Best Friends Pet Hotel which is a 3rd party kennel facility. They are very nice and have a great set of features and facilities. I'd look up their website for more information
Best Friends is just down the street from the campground. They have overnight care and day camps too.
There is a dog daycare but it is not on the Fort Wilderness grounds. Its name is Best Friends Pet Hotel. It is very close to Fort Wilderness.
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