PlayStation and Xbox do not include official drivers, software development kits (SDKs), or operating system support for the Vulkan graphics API.
PlayStation consoles use Sony's proprietary APIs (such as GNM/GNM+), while Xbox consoles use Microsoft's Direct3D (DX12).
Game developers who use Vulkan for PC, Linux, or the Nintendo Switch must write a separate rendering backend or port their game's graphics code to use PlayStation's or Xbox's native APIs before releasing on those consoles.