WASHINGTON—The U.S. House of Representatives voted to repeal an unpopular tax-reporting requirement affecting small business owners, in what would be the first substantial change to the sweeping health-care package signed into law last year.

In a 314-112 vote, the House approved the measure, but the Senate is unlikely to take up the bill as it is currently drafted, adding further delay to the attempt at repealing the measure.

Seventy-six...