The next frontier for the "downloads - PS2 memory card save files - emulator" ecosystem is . Projects like RetroArch (which can run the PCSX2 core via Libretro) now offer built-in cloud saving via Google Drive or Dropbox.
Find your desired save on GameFAQs. Download the .zip file. Inside will likely be a .max file. Extract it to your desktop. Downloads - PS2 Memory Card Save Files -Emulator- - The Tech