apologies if I'm asking something that is described in the documentation, but i could not find it there.
Is there a way to sync your public search keys library in Kleopatra across multiple PCs? Either via a sync-function or for example by syncing the folder in which the keys are stored?
No. And this is out of scope for Kleopatra. You can use existing file sync tools to sync the files in ~/.gnupg. Which files to sync depends on what you want to sync. For details, I suggest to ask on the gnupg-users mailing list.