Depends what sort of sick. I've seen a performer with Montezuma's Revenge get through a gig with humour and "here we go again, talk among yourselves", and be roundly applauded for it. The compere took up the slack.
A fingerpicker with a broken finger managed by flatpicking, not very expertly, and a couple of apologies.
Chest problems can be sometimes got around with monologues or poetry to ease the strain.
Don't think you could manage with migrane. The advice above about having an arrangement with suitable substitutes sounds a good idea to me.