Id go a step further with the cleaning process since you really want the shutters prepared well. Try using 1 part bleach to 5 parts water and a brush. This should do the trick avoid 409 or fantastic, I have heard. good luck

Depending on the type of plastic used, some types of vinyl shutters simply will not hold paint. However, if they have weathered they may have enough texture to hold a coating. You should indeed prime (BM Fresh Start 046 works well) after a thorough cleaning and rinsing. RS exterior should work fine.