Hello!
First make sure to create a backup of all your information, just go to ¨Tools¨ and then click on ¨backup Data¨ select where to back it up and that is it. Then uninstall the program from the old Mac to avoid blocking the license. When you are done make sure you download and install on the new Mac the version of Express Invoice that is compatible with your license or you will not be able to register the license. If you are not sure where to download that from then contact NCH Technical support for them to provide you the download link.
Once you have the program installed on the new Mac, open it up and register the license with the same license registration code you already have.
If you need to contact NCH support you can create a support ticket for them on the link below.
https://www.nch.com.au/support/index.html 
Good luck!