Republicans are pushing forward with a formal investigation even though their yearlong scrutiny of the president and his family has turned up no proof of high crimes or misdemeanors.

The House voted on Wednesday to formally open an impeachment inquiry into President Biden, pushing forward with a yearlong G.O.P. investigation that has failed to produce evidence of anything approaching high crimes or misdemeanors.

Republicans said the vote was needed to give them full authority to continue carrying out their investigation amid anticipated legal challenges from the White House. Democrats have denounced the inquiry as a fishing expedition and a political stunt.

G.O.P. leaders refrained for months from calling a vote to open an impeachment inquiry, given the reservations of mainstream Republicans, many of them from politically competitive districts, about moving forward without proof that Mr. Biden had done anything wrong. Instead, Kevin McCarthy, the speaker at the time, unilaterally announced one in September as he was facing pressure from the far right to deliver on its priorities, including impeaching the president.

But Wednesday’s vote underscored how the political ground has shifted, with Republicans willing to endorse an inquiry even as some emphasized that they were not yet ready to charge the president. The vote was along party lines with all Republicans voting to approve the inquiry and all Democrats opposed.