Sorry, but HP keeps no records of individual Windows product keys associated with individual PCs. So, HP has no way of providing you the product key information.
What you want to do CAN be done, but it will require some considerable effort on your part because it will involve the following: 1) Finding another PC to be used to download stuff 2) Using the other PC to download an ISO file of a repair utility and burning a DVD or creating a USB stick from that 3) On the other PC, downloading a utility that can be used to retrieve the product key from the UEFI firmware of your PC 4) Booting your PC using the repair utility 5) Using a USB stick to transfer the product key utility to your PC and then running it.
You want to download the file Win10586PESE_x64 ISO v160707 file.
Once you have this, you have a choice of media to create: 1) If you can boot from CD, download and install ImgBurn and use the Write Image to Disk option to create a bootable CD. 2) If you can boot from USB, download and install RUFUS and use the option to create a bootable USB stick from the ISO file.