mine had the same issue. took it to this guy who fixes door looks and windows and he took a total of 10 minutes to insert small pieces of foam in a couple of places in all 4 doors. and the vibrating sound's gone.
that's not a problem. try driving any other a/t car and u'll know for sure.
hmm... never heard of this one before. the closest to this i know is that vitz doesn't engage o/d until the engine is warm enough and transmission is disengaged when there's no power (e.g. when car stalls or engine is simply off).
so if your problem only happens on cold engine... and doesn't happen otherwise at all, it probably is an issue with the processor/controller. better get it checked by a very good place! hope it gets solved.