Have you tried contacting people like Europoint to see if there is a substrate suitable for this sort of thing? Most of them cut board to size so shouldn’t be a problem getting the board supplied cut to whatever size you want for you to apply your vinyl prints.
I might be inclined to try a different approach though and try contacting a company that supply printed mouse mats, the ridged sort of mouse mats would probably be ideal for this sort of thing and as the graphic is sealed within the mat then there won’t be a problem with it coming off and should be OK with all the washing.