I can't picture what your strap looks like. As one poster mentioned there may be some caps on the end of the strap you need to pop off. If it don't look like a cap that pops off then maybe the whole piece slides off expousing the screw securing the strap to the door. I bought some aftermarket straps once and the chrome end caps slid over the metal srarp ends.
Chances are the plastic anchor the screw goes into is stripped out or it cracked in half. Thus making it appear to be a loose screw.
The anchor is similar to the license plate screw anchors. If you need a new anchor I think you'll have to remove the entire door panel.
One other thought is the strap may have a threaded stud on each end that goes through the holes in the door panel and is held on by an inverted stamped nut screwed in from the backside of the door panel.
My Bonneville uses this method for attaching the armrest / pullhandle assembly to the door panel.
========= Harryface =========
1991 Pontiac Bonneville LE ~_~_~270,000 miles_~_~_