Yes. The Vita will be backwards compatible with all downloadable PSP games. Next time, please use this little website called "Google". I don't know if you've heard of it. Most people haven't.
While backwards compatibility has been mentioned by Sony, there has been no mention of the % of compatibility. In the past emulators were not 100% compatible with the whole back catalogue. Even the PS2 BC in the first PS3 was only 98%. Even though technology has improved since then I would expect some problems.