WASHINGTON—Companies asking for exemptions from the Trump administration’s tariffs on Chinese imports say widespread shortages of hand sanitizer, disinfectants and other products needed to combat the spread of the coronavirus are being exacerbated by the levies.

Medical-supply companies and other importers have filed dozens of requests for tariff relief in recent weeks, public filings show. These businesses say the levies are increasing the costs of products needed to fight the pandemic and are, in some cases, contributing...