Somewhere along a roughly 7,500-mile journey that begins in Shenzhen, China, there are 19 shipments bound for Rick Woldenberg, the chief executive of Learning Resources, an educational toy company in Vernon Hills, Ill.

Eventually, the containers of puzzle cards, child binoculars and other products will reach a port in the United States, and Mr. Woldenberg will face a difficult and expensive decision. He can pay the sky-high tariffs that President Trump has imposed on most foreign goods, or forgo at least some of the much-needed inventory, perhaps imperiling his bottom line.

Mr. Woldenberg expects to do a bit of both. But he has also opted for a more aggressive course of action, joining a growing roster of opponents now legally challenging Mr. Trump’s ability to issue some of the tariffs in the first place.

Nearly four weeks into a costly global trade war with no end in sight, Mr. Trump is facing a barrage of lawsuits from state officials, small businesses and even once-allied political groups, all contending that the president cannot sidestep Congress and tax virtually any import at levels to his liking.

The lawsuits carry great significance, not just because the tariffs have roiled financial markets and threatened to plunge the United States into a recession. The legal challenges also stand to test Mr. Trump’s claims of expansive presidential power, while illustrating the difficult calculation that his opponents face in deciding whether to fight back and risk retribution.

None of the lawsuits filed this month are supported by major business lobbying groups, even though many organizations — including the U.S. Chamber of Commerce and the Business Roundtable — have been sharply critical of the president’s tariffs and lobbied to lessen their impact. The chamber privately debated bringing a lawsuit, but ultimately decided it was “not the best course of action at this time,” said Neil Bradley, the executive vice president of the group.