What do I need to do to split a washing machine hookup to allow adding a dishwasher?
In the US, a general contractor would do it, sub-contracting all the plumbing work to a licensed plumber.  If you can do all the work that doesn't involve plumbing, you can call around and get estimates, and choose a plumber you like.

From the photo, it looks like you will need to make the support stronger, keeping in mind that the washer is full of water and its center of mass shifts rapidly during operation.  You will also need to robustly attach the washer to the support so it doesn't vibrate off and fall to the floor.