Hector Martin shared that he's managed to get Vulkan working with the Linux 4.10-rc2 kernel and latest AMDGPU-PRO 16.50 driver. He ended up getting the Bonaire firmware to work with the PS4. See the earlier PS4 Linux/Radeon article for more details on the initial bring-up.
More details via this Tweet. In a follow-up tweet he also mentioned that the Vulkan performance on the PS4 was CPU-limited. If you are looking for a game console with official Vulkan API support, you'll want to wait for the Nintendo Switch.