Like Sony has stated at E3, the next Vita firmware update will feature PS One Classics support. The update will be version 1.8. There is no exactly release date stated, but it is supposed to be due out sometime this summer.
Before you get too happy, I have to tell you that not every PS One Classic will be playable right off the bat. You will have to wait until Sony rolls out support for each game before you can play it.
The good side here is that if you have already bought (or buy) the game, you can still play it across the PSP, PS3, and the PS Vita without having to pay any extra.*
*make sure you keep up with your activated systems, you still have to obey Sony's activated systems rule.