There are family restrooms in all of the parks and Downtown Disney. They are oversize and are very private. They are noted on the maps very clearly.
As Cheshire Figment noted, the Companion Restrooms are not marked on the park maps. They are set up for use for people with disabilities (i.e. raised seat toilets and grab bars) and many don't have a changing table, so would not meet the needs of families. There are only 5-6 in each park - which may seem like a lot if you are using them by choice, but not if they are the only toilets you can use.
Some of the handicapped stalls also have a sink within the same area, which would also give you the space and the water needed to cath as hygienic as possible.
The handicapped stalls in AK and the Studio restrooms mostly have a sink right in the stall. The ones in Epcot and MK are older and tend to be just a toilet in the stall.
The restrooms with a sink in the stall and also the Companion Restrooms usually only have a wall hung sink with no countertop. So, plan what you will do for a surface to lay your supplies out.
One thing that might work would be to bring a small rigid cutting board that you can lay across the corner of the sink to put your supplies on. It would be easy to clean and you would not have to worry about finding a surface.
I know First Aid at DLR has private bathrooms they would gladly let you use (and VERY VERY clean BTW), I would assume WDW First Aid is the same, if they were close to where you were at that moment.
WDW also has private bathrooms in First Aid in each park. Again, they only have a sink without a countertop and no real space to lay anything down. You could use one of the rooms in First Aid (not the bathroom) and drain the urine into a container to dump down the sink.
When you go to First Aid, explain what you need.
When you're in the parks, you can get the "Guidebook for Guests with Disabilities" for the park you're in by going to the customer service folks at the front of the park (same place you'd get a GAC).
Just to explain, a GAC is a Guest Assistance Card.
There is more complete information about GACs in the disABILITIES FAQs thread. You may or may not need one, based on the needs you have. You can read the GAC info in the disABILITIES FAQs to figure out.
If you go to Guest Relations, they can give you a copy of the Guidebook for each of the parks, not just the one you are in (if you ask for a Guidebook, they will usually only give you the one for the park you are at, but they will give you a copy of all the Guidebooks if you ask). As was already mentioned, there is a link to the text of the WDW Guidebooks for Guests with Disabilities in the disABILITIES FAQs thread. It's in post #3.