As long as they are cleaned and sanded well, you shouldn't need a primer. The exterior acrylic paints of today are formulated to adhere just fine. Just make sure you use a high quality one.

I have vinyl shutters that were weathered, and I wanted a different color anyway, and they have held very well.
Good luck.

Not all plastic shutters are created equal. Some are PVC which can be painted with a quality acrylic latex topcoat and proper prep. Some are polypropylene which need a specialty primer called Plastic & Vinyl. Some might be polyethylene which can't be painted because polyethylene contains waxes and recyled content that acts as a release agent for paint.