I've been using the OpenVR Input Plugin for years now, and have not seen anything like this. That said, I haven't played Skyrim specifically, yet, I have owned it since release though...
The capture plugin pipes into the OpenVR API and graphs the headset render frame, it should in theory not affect anything at all that has to do with input.
It's interesting that it is when they record or capture, not with OBS simply open, and it doesn't matter if they capture the game mirror, oculus mirror or openvr output.
I would ask to disconnect any other USB input devices, if they have a wheel or cabled gamepad plugged in. I mean, it's still super weird that OBS would cause any of this, but considering Skyrim is a ported title it likely has input hooks for a plethora of other devices. Maybe OBS emulates something? Or something that tries to talk to OBS send it to Skyrim instead.