When you purchased Vegas Pro 21 (assuming from MAGIX), you would have had to have created an account with Magix. This means that you can go to the My Products lisiting under your MAGIX My Account where you will find your Vegas Pro 21 listed as well as its serial number and a download link.
If you purchased Vegas Pro 21 from MAGIX, it will work. Have you given it a try?
Otherwise, you can contact MAGIX Support via the link at the bottom of the Support webpage and give them as much information as possible including the email address that you used to make the purchase, date of purchase, order number, etc.
Please note that this is a user forum and no-one on the forum has access to serial numbers.
If you log into your Magix account, which you had to create when you purchased the software, all you have to do is log into your account, Registered products, and presto.... your serial number. Even if you uninstall the software and reinstall it, presto... the serial number will be there for activation of Vegas.
If you still don't find the Serial number in your MAGIX account > My products, try "Let's Chat" (advice here) and have your relevant email address ready.