NEW HYDE PARK, N.Y.—The Sears store on Thanksgiving night looked as if it was hit by a storm. Shirts, pants, shoes, hangers, empty boxes and tags all were sprawled throughout the store’s three floors at the Lake Success Shopping Center.

But the mess wasn’t the result of a crush of Black Friday bargain hunters. Rather, it’s all that’s left of a once-dominant business at dozens of Sears stores that are being liquidated this holiday season, following the chain’s bankruptcy filing in October.

“I...