Wash cold and hang to dry.
Even better, hand wash cold and hang to dry.
Vinyl is a tricky beast. If I gets certain compounds on it, it'll break down. Same with heat.
Warning: I do not have one of these changing mats or anything vinyl, but I have read several others warn about vinyl breaking down. I also work regularly with vintage plush toys and hand-wash is what I do with them. They come out all nice and clean. And by hand-washing it will allow you do see if you are doing any sort of damage to the item and stop before it's completely ruined.
Edit: Hand-washing is not the same thing as 'surface-washing'. :laugh: Hand washing needs a sink/bucket/bathtub and you just put the detergent on and smoosh away until it's clean. Sometimes I bring an old toothbrush to the party to get at stubborn spots.
A surface wash is where you get a sponge or cloth damp and run in over the item in question. Like a plush toy with electrical components.