To answer the direct question, I don't think there is a simple answer that either metal is better or wood is better. Figuring that out probably requires to know the frequency of the vibration, the dimensions of the legs, the weight supported, the type of wood or metal, and other things.
But no matter. To solving your actual problem, I endorse niels nielson's recommendation to use a box spring, and also add to your bed a layer of memory foam type material, perhaps a mattress pad.
Also, go ahead and get that frame. Whether it's wood or metal I think it would help.
Is the mattress/frame/springs/headboard touching the wall? If yes, pull it out a bit. If no, shove it firmly up against the wall. Who knows, you might change the harmonics or the natural frequency of the bed system and reduce vibrations.
You mentioned that the opener is old. Is it a chain drive? Perhaps a bit of light chain maintenance would smooth up the operation a bit. You could lube it up with some cheap bicycle chain lube.