Any way to mount this diffuser without screws?
Nothing should ever be mounted just to the drywall. Whenever possible, everything should be mounted directly to the substructure (studs,blocking, etc.) When framing is not available hollow wall anchors should be used, but only in light duty applications. In the case of your vent I would suspect that two sides abut the rafters or joists. Locate them and try to run the mounting screws for your diffuser through the drywall into the structure. If this is not possible, mount some 1x material to the inside of your duct with sheet metal screws and secure the diffuser to that.