Doing it without a vote certainly looks weak. From the NYT article:
>>>The apparent decision not to seek a vote was a tacit acknowledgment by Mr. McCarthy that he lacks the numbers to do so amid G.O.P. divisions. Several Republicans, including those from districts Mr. Biden won, have indicated they did not support an impeachment inquiry unless investigators could tie the business dealings of Hunter Biden, the president’s son who engaged in transactions with overseas firms, to his father, or uncover evidence of high crimes and misdemeanors.<<<
It's not clear that anyone in the senate is in favor of this:
Senate Republicans say the House GOP doesn’t appear to have enough evidence to pursue impeachment proceedings against President Biden and are skeptical about the prospect of setting up an inquiry w…
thehill.com