Don't have any with Steam controller either.
Now works properly with recent Steam Beta Update.
Sounds like this is resolved, then. I just started playing yesterday (August 2025) with vanilla Proton 9.0-4, PS3 controller has vibrations. (Unrelated: XBox button labels are confusing, however, but this seems to be a common issue with PC games assuming XBox system if using a controller.)
Closing per the last couple comments.
proton 9.0-4x1 2025-08
I use DS3 gamepad. Vibrations doesn't work in NieR:Automata, but it works in other Windows games like Mirror's Edge (17410), SAO FB (626690), The Legend of Korra (281690).
OS: Manjaro Unstable (Kernel 4.18.5),
GPU: GTX 1080 (396.54)
CPU: i7-8700k
Proton: 3.7-5
Edit: Some vibrations work, but on Windows I have more vibrations.