The partial U.S. government shutdown has docked fishing boats in Alaska, delayed public meetings on a proposed wind farm off the Massachusetts coast, and blocked pharmaceutical companies from seeking approval for new drugs.

But the Trump administration is working overtime to make sure the shutdown doesn’t halt oil drilling too—in ways critics said could flout federal law.

“One of the principles of government is that you serve everybody equally,” but that isn’t what is happening here, said Matt Lee-Ashley, a former deputy chief of staff at the Interior Department. “The oil industry is still getting business as usual and everybody...