The incipient Biden campaign has raced to lock down the party’s biggest donors in recent weeks, pressing the message in private calls that the former vice president’s ability to marshal funds quickly will represent the first major test of his run, according to people who have been contacted. Mr. Biden’s allies have pointed with concern to the $6 million sums that Mr. Sanders and former Representative Beto O’Rourke generated in their campaigns’ first 24 hours as the high bar against which he will be measured.

Unlike those two rivals, Mr. Biden does not have an at-the-ready list of hundreds of thousands of contributors to ply for small donations. He must rely heavily, at least at first, upon an old-fashioned network of money bundlers — political insiders, former ambassadors and business executives who can expedite dozens, if not hundreds, of checks for $2,800 each, the legal maximum an individual can contribute in the primary.

But there is an inherent tension in the pursuit of big money in the current Democratic Party: the more of it that Mr. Biden gobbles up, the greater the risk of a backlash from a liberal base skeptical of the influence of the wealthy on the party.