Specifically, those purchased legally through the PSN Store. It seems like it should, but I can't find a definitive answer anywhere.
Update 1: PSOne Classics will come to the Vita August 28, 2012 (Joystiq)
Update 2: On August 28, PlayStation Vita firmware update 1.8 went live, enabling the Vita to play select PSOne Classics. IGN has a list of compatible PSOne games.
Update 3: At this point most, but not all, PSX classics are supported on Vita. As per a recent mixup, all PSX games were temporarily downloadable on PS Vita, and all of them work. The reason any remaining titles aren't available on Vita is likely due to licensing issues.
PS Vita didn't support PSOne/archive titles at launch. You will need to wait.
Japanese PS Vita FAQ page reveals upcoming handheld will not play archived titles initially, says more details will be revealed later.
And from the Playstation Blog:
Q: Is PS Vita compatible with other PlayStation platform games like PS3, PS2, or PSP? Do you have any plan to support other PlayStation platforms in the future?
A: Users can play supported digital PSP titles and supported minis. In regard to PSone classics titles, we will announce further details when they are ready. PlayStation does not have plans to make PS Vita compatible with PS3 or PS2 titles at this point.
At this time, PSone titles are not supported by the Vita, along with the Turbografix and PS2 titles. According to some google translated pages from Japan, the PCEngine/Turbografix and PSone titles support will be finalized/announced at some point in the future. I imagine the only reason it's not supported yet is that they need to finish the emulator, as it should be a simple matter to run them on the vita hardware.
Whilst not helpful if you want to play whilst out and about, I believe you can play PS1 games using remote play through a PS3.