Drilling under a vent
Looks like a return cover, which means there's likely no ducting at all (just an open stud bay).  

If that's the case, putting a nail/screw/anchor in the cavity, could lead to dust and debris getting snagged up on the object.  Will it cause a problem? Probably not, but it surely could.

Since you're  looking to hang a coat rack, you should do your best to attach it directly to framing members (studs).  Coat racks can be loaded with quite a bit of weight, and are often subject to abuse (yanking, pulling, etc.).