TOKYO—A representative from Apple Inc. recently called Kureha Corp.'s offices in the U.S. The problem: Apple was facing tight supplies of lithium-ion batteries used in its popular iPods, and they traced the supply bottleneck to the relatively obscure Japanese chemicals maker.

Kureha, which has a 70% share of the global market for a crucial polymer used in lithium-ion batteries, had to shut its factory in Iwaki—near the quake's epicenter—after the March 11 disaster struck. It is the only place where Kureha makes this particular...