Yet perhaps more than ever, Mr. Sanders is betting on the grass-roots appeal that propelled his 2016 campaign, a factor that his campaign says polls often fail to capture.

Rather than relying on high-dollar fund-raisers — events he categorically rejects — he instead hopes to energize enthusiastic supporters who can donate to him again and again. In a show of force last month, his campaign said it had logged contributions from more than one million individual donors. His campaign said the average donation for the quarter was $18.07.

His strategy means he does not have to worry about donors maxing out; his campaign said more than 99.9 percent can give again. But it also poses challenges: Though he has a loyal army of supporters and volunteers, many of them do not donate large amounts of money, meaning he must appeal to a huge pool of donors in order to keep pace with rivals like Mr. Buttigieg, who is willing to collect large sums of money on the traditional big donor fund-raising circuit.