The next Vita firmware update will add the PSOne Classics support originally announced during E3. There's no exact date for version 1.8, but it's due sometime this summer.
The update won't simply enable all PSOne Classics; according to Sony representatives, support for PSOne games will be rolled out a few at a timefollowing the new firmware. The good news is you'll just have to buy (or have already bought) each supported game once for use on PS3, PSP, or Vita.