The vent typically would screw to a flange at the end of the duct work, meaning the screw just go thru the drywall. Those should definitely be repaired. Also is that dark stuff on the drywall edges mildew? If so you definitely need to fix that before moving forward as well.

Is that a layer of plywood above the sheet rock? If it is then use 1 inch wood or machine thread screw so it can grab the plywood.

My vents have a flat 2x4 at each end up between the floor joists, and the screws for the vent cover go through the drywall and into the wood. I have about 1 1 /4" long screws and they penetrate the 1/2" drywall and get plenty of bite into the wood. If you have plywood up there it would serve the same purpose. It doesn't take much to hold those covers in place.

Bruce

Thank you all for your comments. I think that is mildew on the drywall, so I'll need to address that as well.