Instead of starting chimerax with the Steam run.sh script it is possible to add to the SteamVR vrclient.so library a path that it needs to find libvulkan.so.1. I found (using ldd) that vrclient.so depends only on libvulkan.so.1 from Steam and specifying the location of libvulkan is sufficient for ChimeraX to run VR, starting Chimera with LD_LIBRARY_PATH=~/.steam/ubuntu12_32/steam-runtime/amd64/usr/lib/x86_64-linux-gnu chimerax-daily. This path can instead be added to the SteamVR vrclient.so library that the Python OpenVR library used by ChimeraX imports.
sudo apt-get install patchelf
cd ~/.steam/steam/steamapps/common/SteamVR/bin/linux64
cp -p vrclient.so vrclient.so.orig
patchelf --set-rpath '$ORIGIN/../../../../../../ubuntu12_32/steam-runtime/amd64/usr/lib/x86_64-linux-gnu' vrclient.so
This will break when SteamVR updates its vrclient.so library.