What is the value of a bicentennial quarter that is missing the nickel plating on the face side and is thinner and lighter than a regular quarter?

What you have is most likely a "lamination error". If so, it could retail for about $10 but it would need to be evaluated in person by a dealer or appraiser who works with error coins.

Current US dimes, quarters, and half dollars have two outer layers of cupronickel cladding* bonded to a copper core. Blanks for the coins are punched out of large rolls of metal stock. Sometimes the ends of the rolls aren't properly clad and one or both sides may be missing from the copper core.

* "Plating" is normally a thin coat of metal applied electrically or chemically in a liquid bath. Cladding for coins is much thicker and is bonded mechanically.