ANNAPOLIS, Md. — Larry Hogan was annoyed. On a conference call, Mr. Hogan, the Republican governor of Maryland, had just learned that several South Korean companies were ready to ship more coronavirus test kits to his state. But they were stymied because the Food and Drug Administration had not yet approved their use.

“I don’t care if we have F.D.A. approval or not,” Mr. Hogan said into a speakerphone in the governor’s reception room, where he was flanked by a container of Purell and a 9 a.m. Diet Coke, with aides sitting six feet apart around a large table. “We’ve got people dying,” he said, adding, “I don’t want to wait for permission.”

Frustrated by limited support and unclear guidance from the Trump administration, governors across the country, including some Republicans, have been squaring off with the White House and striking out on their own to secure supplies. Mr. Hogan, in his second term in a very blue state, has tried to stay miles ahead of the virus’s incursion here, like several other governors — notably Jay Inslee of Washington and Mike DeWine of Ohio — whose responses have been given better marks from Americans than the president’s.