On the assumption that you bought Vegas Pro 21 Edit from the official Magix Vegas website, check to make sure that the purchase email hasn't gone to your spam/junk folder. Otherwise, you would have to have created a MAGIX account in order to make the purchase. Therefore, login to your MAGIX My Account and then go to the My Products listing where you will find Vegas Pro 21 details including its serial number.
It's not a good idea to post private information such as order numbers on the forum - this is a user-to-user forum, not MAGIX Support.
Your MAGIX account link here. If you don't find anything as advised above, try emailing shop@magix.net and explain your issue including your order number.