Among the prominent figures to be subpoenaed by the Democrats are Jeff Sessions, the former attorney general; Rod J. Rosenstein, his deputy who appointed Mr. Mueller, the special counsel; John F. Kelly, the former White House chief of staff; Jared Kushner, the president’s son-in-law and senior adviser; and Corey Lewandowski, a former Trump campaign manager. Democrats also authorized a subpoena for David J. Pecker, who as head of American Media helped Mr. Trump during the 2016 presidential campaign buy the silence of a pornographic film actress and a former Playboy model, both of whom claimed to have had sexual relationships with him.

The committee’s actions set up a slew of possible new conflicts with the White House, which has taken a dim view of House Democrats’ continued investigation of matters studied by Mr. Mueller. White House officials could try to intervene to block testimony from many of those subpoenaed on Thursday who are current or former high-level administration officials, as they have with other witnesses.

That would only deepen the standoff between the administration and the House. On the census issue, Mr. Barr and Mr. Ross could still reach an accommodation with the House Oversight and Reform Committee, but more likely, Tuesday’s vote would allow the committee to go to court to try to pry the documents loose and make criminal referrals for Mr. Barr and Mr. Ross to the Justice Department for defying congressional subpoenas.

Despite rancorous Republican opposition, Judiciary Committee Democrats easily pushed the subpoena authorizations through on Thursday along party lines — promising to jump-start two of their highest-priority oversight investigations of Mr. Trump and his presidency. The chairman, Representative Jerrold Nadler of New York, did not indicate when he would deploy the newly authorized orders, but he is likely to wait until after closely anticipated testimony in the committee next week from Mr. Mueller himself.

The first oversight inquiry focuses on Mr. Trump’s attempts to impede federal investigators studying his campaign’s ties to Russia constituted obstruction of justice or an abuse of power.