How to fasten wooden board and batten shutters to vinyl siding
Typically, #10 stainless steel screws.  However, it depends on what is behind your siding and what material was used to construct the shutters.  Cedar or pine is very common in shutters and is light which presents no problem.  I have seen shutters constructed of composite materials that can be fairly heavy and thus, you may want to go to a #12 and/or add more of them.  More importantly, they need to be anchored into something solid behind the vinyl siding.  If your wall sheathing is plywood or OSB you should not have a problem.  If it is insulation board/blackboard or something along those lines, you will need to either hit a stud or install mounting blocks.  Also, you will want to pre-drill all your holes, fill the holes in the vinyl with caulk or similar sealant, then drive your screws to hedge against water intrusion.  Hope this helps.