Web the simple answer to the question is no, you can’t retake the quiz if you’ve already taken it on that account. They say the wand chooses the wizard, but there are definitely ways around that. There have been some claims in the past that.
Wizarding world wand quiz all questions
My wizarding world patronus quiz
Harry potter wizarding world wand quiz
Wizarding World Account reset to to retake house and patronus quizzes
You can retake the wand ceremony in the harry potter fan.