Basically, just install Vegas Pro (14 +) on the new computer and if both activations slots have been used, go to My Products on My Account on MAGIX and de-activate one or both of those activations (only once per month allowed). Or go to your old computer and de-activate from the Help menu.
You have a valid serial number for the software. You can download the software e.g. from your MAGIX account or from the MAGIX Download center. Install and activate as before. Worth also checking out point 9 in the link given.