Alen Ladavac of Croteam has revealed in a Steam Community thread they intend to support Vulkan with Serious Sam 4, Talos Principle, and potentially Serious Sam 3.
Serious Sam 4 remains under development and "AlenL" commented, "Yes we are working on it [Serious Sam 4 and Linux]. But good things take time. ;)" When prompted whether Vulkan will be utilized, he responded, "We will support Vulkan. But we probably won't be removing support for other APIs soon." So Vulkan should be supported alongside other APIs, which would make for an interesting comparison of OpenGL vs. Vulkan performance assuming the games benchmark well in an automated manner. This is also obviously good news for everyone since the older graphics cards / drivers will not receive Vulkan support.
When prompted about other Croteam games in that thread, Ladavac responded, "SS4 certainly will not use the same version of engine as Talos does. It will use a _much_ improved version. Exact details will be revealed when SS4 is officially presented. Talos will support Vulkan on Launch. SS3 not yet, but maybe later." So The Talos Principle is getting Vulkan support at launch while Serious Sam 3 may be supported later.
Serious Sam 3 is powered by Croteam's Serious Engine 3.5 while Talos Principle is powered by Serious Engine 4. Croteam isn't a Khronos Group member but sounds like they've already got access to the new graphics API or intend to support it quite quickly once the specification has been ratified within the next few months.