Simple, non-retractable faired wheels (e.g. C172, Grumman Cheetah)
can rotate in flight. That doesn't mean they always do.
The bottom of the wheel sticks out of the fairing into the slipstream, and thus can be spun by the passing air, like a stream waterwheel.
https://en.wikipedia.org/wiki/Water_wheel#/media/File:Stream_waterwheel_simple.svg
Mass of the wheel, axle lubrication, roughness of the tire tread, and aircraft speed will all affect whether the force is enough to spin the wheel.
I've never felt vibrations, myself.
Complex wheels that are faired into the fuselage once retracted (e.g. 737) won't have the same asymmetrical force, and usually are braked to a standstill during the retraction process.