The bench needs to be rigid, in any case. So brace the back and ends, and reinforce the top, as already advised; at least 15 mm ply (Chipboard might fragment in the future)
Attaching it to the shed walls will allow any noise to “telephone” through, and be more noticeable.
Having made the bench as rigid as possible, look at the belts.
Excessive belt tension can overload, and wear, bearings in the motor, countershaft and spindle.
You need to have enough to allow a little play , halfway between pulleys.
This can be a safety feature; if there is a jam up, the belt can slip, rather bthan stall mthe motor, or damage something.
If the belt has stiffened through age, or has a prominent join, that can cause vibration.
Sometimes fitting a new belt, or link type belt can reduce or banish the vibration and noise.
If you are using changewheels, set with a little backlash.Setting with a sheet of paper in each mesh should suffice.
First set the backlash between Leadscrew gear and Idler.
The swing the banjo to set the backlash between Idler and Spindle gear.
If you are using a compound Idler, set the backlsh between each mesh in the same way.
HTH
Howard