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I haven't had a PS4 for a few months now, but if I recall correctly I don't think there is a "classics" section (yet) on the PS4 store and Vita games aren't shown on that end.
Whenever I made PSN purchases I would use the online store anyway, store.playstation.com ... Much easier to deal with and you can usually start a remote download after purchase to whatever system.
Like I said I switched to PC and I don't know if there have been major changes since then. Good luck!