This issue is precisely what the Web Department has been set up for. Many skilled designers can’t code, or don’t have time to learn and then keep up to date with current best practice in coding. After all, it changes almost daily. Many designers want to concentrate on design trends.
So, with some technical backup, the answer is yes. You can be a web designer without knowing how to code – just get the technical back up that you and your clients deserve to ensure your sites have optimum functionality.