Roughly 80% of Iridium Communications Inc.’s 1990s-vintage satellites are operating without in-orbit spares, further raising the stakes for launching replacements on a Space Exploration Technologies Corp. Falcon 9 rocket slated for the fall.

With two of its remaining 66 core satellites out of operation after nearly two decades—and a single backup satellite available for only a small portion of the reduced constellation—Iridium is betting on the coming launch of 10 next-generation spacecraft scheduled between mid-September...