To save myself from running around during feeding time, I secure bibs on the back of the highchair so that they are ready to go. I also string a small laundry bag for the dirty ones. Most high chairs have a bar in the back that allow you to open the bib and secure it there.
This page contains the following solutions.
I just read in the newspaper that some vinyl bibs from Toys R Us were being recalled because some of them had too much lead. Made in China. It's possible that some of the bibs ended up at other stores under different labels.