The fiber glass is glued onto 1/8" or 1/4" plywood. If the wood chips out and there's no framing where you need to put the screw, Lowe's sells metal wall anchors that you put in the hole and tighten the screw to crimp the anchor into the wall. You then remove the screw and put the vent back in with the screw that came with the anchor. How well they holds depends on the condition of the rest of the plywood around the hole.
There are different lengths for different thicknesses of wall.
I've also removed a loose fixture, carved out the foam insulation, and installed the appropriate sized wood framing around the opening to give me something to screw to. That takes more effort but is probably the best method as it will probably last longer.
