You can download your version here:
If you have registered you license of Lightroom you'll find the serial number if you logged in with your Adobe ID.
I was able to download the software but when I log into Adobe, I don't see anything in the purchase or order history. I must not have registered it. Any other suggestions? I have my order # in an email from when I purchased it.
Without the serial number, it would seem that your only choice would be to purchase Lightroom 6 as a new customer. That would mean you would have to pay full price because in order to purchase an upgrade you need the previous serial number. If you have your old computer still there is a way to retrieve that serial number. What operating system did you use?
Another longshot would be to do a live chat session with Adobe and see if they can help you.
What version of Lightroom was that? In early 2009, the version was 2.3 or so. Even fully updated (which will only get you to 2.7) It will likely not run on a current operating system and hardware so you probably have to upgrade anyway for your new machine. You either do the $10 a month CC subscription for Lightroom+Photoshop or buy the standalone LR 6.
Even if you can get that very old version to install, and you probably cannot, there is a good chance that the registration servers needed to support that very old version are no longer available.
Since you have an new computer running a current operating system, you will be very happy with the new version of Lightroom that you should upgrade to.