To this time there were 2 Webkit Exploits released for the Vita and PS4. The first one (CVE-2012-3748) for 3.20 and below and the second (CVE-2014-1303) for 3.36 and below. Sadly both got patched in the following updates by Sony.
Test the old PoC for <3.20 now: lolhax.org/vita
vitasploit-33x.zip by xyz
VitaSploit by Hykem
Pong Sample by Barnabe42 (for <3.20)
Frequently Asked Questions
What if I'm on 3.50 or any higher Firmware?
Sorry, as Sony updated the WebKit Software this is for 3.36 and lower only.
When will we have native homebrews?
Check out the PSM hack called Rejuvenate for that!