Welcome to Adobe Forum,
Since the version of Contribute that you have is not known, please check the technical compatibilty of the version of Contribute with your new machine http://www.adobe.com/products/contribute/tech-specs.html,
if found compatible then follow this :
Hope it helps you.
Ver 6.5 single user
If that is not a CLOUD version, you install the same way you did the first time... put your install disc (CD or DVD) in the drive and follow the instructions on the screen
If that IS a Cloud version, you log in to your paid Cloud subscription and install from there
This was download from Adobe.com