Steel and aluminum importers that obtained the earliest Commerce Department tariff exemptions are scrambling to get renewals before the original ones expire in June.

Companies complain the pending deadline adds new burdens to an already backlogged process that’s forced them to prepare months in advance for renewals while still waiting for the Commerce Department to rule on pending applications.

“It’s not as easy as getting your driver’s license renewed,” said Riverdale Mills Corp. Chief Executive Officer Jim Knott Jr. Riverdale, of Northbridge, Mass., got exemptions for only 11 products despite requesting 119 exemptions, he said. The renewal process is creating...