Oscar fanatic James A. Molnar spends a good part of every winter chasing down hard-to-find Academy award nominated films, zipping to theaters around Ohio in his dark blue Mazda with the license plate OSCARS.

“As a kid I just loved the pageantry of the Oscars, and that dovetailed into me loving movies,” says Mr. Molnar, a 31-year-old digital media specialist from Toledo who has two Oscars, a $300 fake one at home and a golden replica tattooed on his right hip. Each year for the past decade, he has attempted to see every Oscar...