The chairman of the House tax-writing committee formally requested President Trump’s tax returns on Wednesday, kicking off what could be a bruising legal fight between Congress and the Trump administration.

In a letter to the Internal Revenue Service, Rep. Richard Neal (D., Mass.) asked for six years of Mr. Trump’s personal returns and some of his business returns, invoking a federal law that allows the chairman of the House Ways and Means Committee to seek any person’s tax information.

Mr....