I would place the plate carrier into a fabric bag or pillow case to give it some protection in the washing machine on cold wash cycle with small amount of detergent.  Add a cup of white vinegar (not malt vinegar used for fish and chips) because the vinegar is gently acidic and will help break down the smoke molecules causing the smell.  As suggested above simply air dry the plate carrier.  Any smell of vinegar will dissipate.   Alternately, do not use the washing machine but a bath/sho