If its air sick and barfing in a bag personally I wouldn't
Any hint of them passing out or becoming unresponsive to verbal instructions declare PAN or MAYDAY it doesn't really matter what you call the reponce will be exactly the same from the ATC.
Also if they start leaking bodly fluids apart from barfing again declare.
But have a think on your own work load. If they are causing you problems and degrading your ability to manage the flight safely call a pan for a barfing pax. Your not an instructor who can form a barf bag using a chart while flying the approach using the rudder pedals and holding some ones hair out of the puke while trying to flick away thier mike.
Personally I wouldn't hand out barf bags at the start of the flight just have them under your knee board or in a door pocket within easy reach. If someone is in a mind they will feel sick a barf bag will reforce that feeling.