Yaakuro, a community developer working on Unreal Engine 4's Linux support, has got the basics of UE4 running with SteamVR's Linux beta. There is currently a small issue with the VR Editor but other aspects should be working, including supported UE4 games.
Currently though Unreal Engine 4 on Linux SteamVR is using OpenGL for rendering rather than Vulkan. UE4's Vulkan support on desktop platforms has been lagging compared to their Vulkan support on Android, but hopefully we'll see their desktop Vulkan support in better shape soon.
Here's a video by Yaakuro showing off UE4 with SteamVR on Linux.