I don't know how NVIDIA's linux driver is structured but I'm curious if they use a separate Vulkan .so file like NVIDIA's OpenGL library which one has to make sure is used instead of Mesa.
The good news is that Vulkan was designed with a multi-GPU (vendor) use case in mind so there is a generic "loader" that you use to load vendor-specific Vulkan implementations for each GPU your application renders with.
When you create a Vulkan context you first need to enumerate the rendering devices on the system and select the one you want to use. You can actually explicitly use multiple contexts for different devices in your application, e.g. using the IGP for compute tasks while doing the main rendering on a dedicated GPU.
Please also note that AMD's Vulkan beta driver is not packaged with DirectX, so you should give it a pass unless you are a dev or you have a game that supports it.