The PlayStation Vita officially supports PSOne Classics games now. But not all of them, at least not officially. As of today, you'll be able to get .... thirteen of them from the PlayStation Store directly through the Vita. You can get many that aren't on the store, if you have previously bought them on PS3 or PSP, by checking your account's download history and downloading from there.
But that doesn't work for every game – many PSOne games are not available for download onto Vita through your history. Many of those can be copied to Vita from PS3 ... if you happen to have them sitting on your PS3 without installing them. And that doesn't cover all games either.