It shouldn't be hard to find a used 24" x 80" hollow core flush door 'slab' (not machined) at a ReStore or salvage store. Even Masonite wold work. The rails and styles are usually ~3/4" thick and a 24" door should be about the right width. If it's machined already just get a wider one. Cut the door with a straightedge and glue a piece of 3/4" pine back into the end and Bob's your uncle.....
The suggestion I would have is this: I assume you have a front loading washer/dryer combo with front controls, so access to the rear of the machine is not needed unless changing machines or a hose bursts. In that case, I would drill 1/4" locator holes in the wall cleats and use an appropriate length smaller screw to register into the hole to prevent it sliding out. You could lift the shelf off quickly if an emergency arises and it would still be stable.
