I am not with Laminar Research; just a flight simmer.
The sad answer is that there is not a product key with a boxed set Disc 1 has to be installed in the drive at all times for x-plane to run.
If you purchased the boxed set from a third party other than Laminar Research then the so called "key" was a promotional number to entice you to purchase additional products from the reseller or its agents.
If the boxed set was purchased directly from Laminar Research then you are asking your original question in the wrong place. You need to contact Laminar for starters at email@example.com but before you do you need to assemble full details of the purchase ie when, how purchased any receipt numbers etc
Good luck otherwise you will be looking towards the full purchase of a digital version.