While you can scrape these links via XML files, the official Support Page is the safest bet for official firmware.
The 3.65 updater expects the PUP file to be placed after you install the updater VPK, as the installation process may delete any existing PUP file in that location. psp2updat pup download link
This article provides verified information on obtaining the official PS Vita system software, instructions on how to use it, and how to safely install it. What is PSP2UPDAT.PUP? While you can scrape these links via XML
Follow the official instructions from TheOfficialFloW: What is PSP2UPDAT
When downloading from community archives, check the MD5 or SHA-1 hash provided by the site to ensure the file is not corrupted.
"The latest version of the system software is already installed"
If you are forcing an older update, you may need an psp2-updatelist.xml file in the same directory to trick the Vita into seeing the file.