I can't figure out what this part is and I need a replacement. I somehow ripped it running over a curb.

2020 F150 King Ranch

this is some kind of mud guard made of fibrous material on the undercarriage behind the metal skid plate.

It seems to cover some kind of intake, so I think I should replace it rather than just leave it off.

It is directly behind this metal skid plate:

Any idea where I can find one of these or what is it called? When I google mud guards I get all kinds of different things.

 

Most likely the belly pan. It encloses the area under the engine and makes air flow under the truck better. It looks to me like you could get that one back on, maybe need a couple of big washers for where it’s broke.

 

That's interesting, I didn't even think of that. I'm a little worried it will vibrate loose though. Maybe I could jb weld it to some bigger washers.

 

Those 'fall off' quite often. Often is an overstatement but you are far from the first or only.
Theory is the oil chance places won't remove all the hardware, yet leave it hang under the vehicle during an oil change weakening one of the corners until it finally falls off.

You can replace it if you want but it's not detrimental.

I would like to keep it installed, just to keep small road debris from hitting a sensor wire or other thing that shouldn't break because a rock hit it. They say too it's meant to be on the 'luxary' vehicles to help reduce noise bounce-back to the cab after echoing off the ground.

 

You don't need it. I'd leave it off.