The 'in-bed' part of your 5th-wheel hitch should fit just fine, the only question is the underbed frame-mount parts (are all of the frame-mounting bolts/holes/clearances exactly identical for 2021 and 2022?)
B&W does have the Turnoverball kit for the 2021 F150 (B&W PN GNRK-1121) but as of this date have not yet proven that for fitment to the 2022. They expect to have an approved 2022 version "soon", I'm on a notification list for when that is released (it may be the same as the 2021 or may be different, that's not yet determined by B&W). The same is true for their "Patriot" rail mounted 5th-wheel kit. The devil is in the details for these 'no-drill' frame-mount systems and B&W takes great care to ensure compatibility for each application.
This may vary depending on exactly which underbed mount system brand and type (5th-wheel rails mount, etc) you seek. I suggest that you contact the manufacturer of whatever underbed frame-mount system is of interest to ensure that the components are a proven fit for your 2022 F150.