What is the program, exactly? Adobe have different things, and the way you use them is very different. If you aren't sure of the name, please tell us what you do, or click, or type, to make it work.
As per the description above, you are moving to a new laptop and want to access the service of Adobe ExportPDF, Is that correct?
The record shows that you are subscribed to Adobe ExportPDF service on your current Adobe ID, you can access the service through the URL https://cloud.acrobat.com/exportpdf with your Adobe ID and password on a browser like Chrome, or Firefox, or Safari.
You can also use the freeware Adobe Reader DC to use the service of Adobe ExportPDF, just "sign-in" with your Adobe ID and password in Adobe Reader DC, to download Adobe Reader DC, use the URL Adobe - Adobe Acrobat Reader DC Distribution
For troubleshooting, you can refer to Troubleshooting Create PDF, Export PDF, Adobe Send, Fill & Sign … services(PDFPack or Document Cloud...
Note: Adobe Document Cloud Online services cannot be downloaded as an application on a computer. But we can access the services through a browser and can be accessed through desktop and mobile app Adobe Reader DC.
Feel free to update this discussion for any further assistance.