Depends what you mean by "design". They get to choose some aspects of how it looks in general, but as for the actual schematics/PCB layout, that's based on the CPU manufacturer's reference design.
Edit: the amount of downvoting the original comment got, and the sibling that got killed for saying effectively the same thing, makes me think someone is trying to push a certain narrative, but the truth is out there. No one except a very small number of OEMs actually designs PC mobos, and they are also based on reference designs from Intel or AMD.