It looks like PSone Classics have finally come to the PlayStation Vita, but watch out! Very few games will actually work if you download them through the PlayStation Store directly from your Vita at this time. Instead, you’re basically going to need to transfer them from your PS3 to your Vita, which sadly means you need a PS3.
In order to do the transfer, download the game through your PS3 and leave it in “bubble” form (uninstalled). Plug in your Vita and go to the Content Manager. Select the game (and hope it is compatible) and transfer it to your Vita. You can also transfer saves this way.
Note that not every game is compatible, so don’t rush off to just buy any PSone Classic assuming it’ll work on your Vita. Full user reported lists can be found here and here, since Sony isn’t content with giving their fans an actual list outside of nine “officially” supported US games (The EU has 100 and Japan has more than that).