Ambriola Co Inc’s mammoth warehouse in West Caldwell, New Jersey, is crammed full of boxes and wheels of harder cheeses such as Parmigiano Reggiano, Pecorino Romano and Grana Padano - and more is coming, lots more.

Phil Marfuggi, president and chief executive officer of Ambriola, a unit of Auricchio SpA, one of Italy’s largest cheese producers, is among the many importers and shop owners across the country who are scrambling to stockpile European cheeses before new U.S. tariffs kick in on Oct. 18 in efforts to shield consumers from price hikes.