Howdy Fishman,
First recommendation is to remove the spare tire carrier so you can see up there. I did this a month or so ago when replacing some u-joints and it hasn't made it back on yet - not sure if it ever will. Anyhow...
Laying under the rear of the car looking up you will see the actual metal bumper hidden behind the fiberglass, plastic, whatever you call it. On top and in the middle of the metal bumper is an aluminum support bracket about 8 to 10 inches long and triangular shaped. The three nuts holding the corvette emblem on are behind that. If you can get a wrench or small hand back there, great - go for it. If not, then you'll have to remove the aluminum triangle. The triangular aluminum support bracket is held on by 3 bolts, one of which you will see centered on the bracket going from the front of the car rearward into the bumper. The other two are inside the triangle pointing down into the bumper. The one screw you can see is a self tapping screw - no problem. The catch is that the other two inside the triangle are not screws, they are bolts with nuts that you can't get at unless you remove the plastic/fiberglass bumper cover. You might be able to get those two off without removing the bumper cover body panels, but you won't get them back on without it. On my car, these two bolts are no longer present. I'm guessing that when a previous owner repainted the car, he figured that the one self tapping screw in the middle was more than enough to hold that aluminum triangle on - and indeed, so far, so good. But I wonder, are the two nuts from the missing bolts still floating around between the metal bumper and the fiberglass cover? I'm hoping that I never need to take the thing apart far enough to find out.
Semper Fidelis,
Culprit