Lionel Post War 3656 Stock Car & Platform. Can I power it with separate transformer, independent of the track?

Yes. Simply take the wire feeding the switch from the track connection and connect it to your accessory power source (this presumes you are following the common ground standard with phased power sources).

Great. Thank you. 

Several years ago I recall an article on this accessory in which the author advocatedtwo separate variable control supplies, one for the corral/platform, one for the car.  His experience said that the 'sweet spot' voltage for each vibrating unit was different for optimum operation.  Could be. 

I acquired one of these sets in (seemingly) excellent condition, but have never taken the time to insert it into the layout.....I've pretty well migrated towards the scale equipment on the layout through the years, leaving the traditional-sized (e.g., 3656 cattle car) cars/engines aside.  Nonetheless, being a mechanical engineer by training, I've always had a soft spot for gizmos and gadgetry of the hobby....such as this combo.  

I suppose it's an accessory that's defied improvement in the Lion Hall of engineering.  They've never made a so-called "remake", incorporating better design/components/materials since their last generation released in the mid-1950's.

Anyhow, I recall Ray Plummer's final paragraph regarding this accessory in his old Kalmbach repair book for Lionel items as being the final answer to balky bovines.  It said something to the effect that, when all else fails, even the youngest railroader would 'get the most stubborn of the herd mooooving with a pinky poke to the rump roast.'    

BTW, good opportunity for a sound module of distraction!....like ITT Products' GL401 "Cattle" sounds.  In fact, pump that chip through a BIG 8-ohm speaker mounted under the accessory....might just provide that 3rd level of 'vibration' that gets everything in harmony!!!

Good luck!

KD