WASHINGTON—Joe Biden faces a distinct funding disadvantage against his major Democratic presidential rivals in the months leading up to contests in Iowa and New Hampshire, creating potential obstacles in his bid to challenge President Trump.

The former vice president, who has never been a prolific fundraiser, has seen his strength in polling undercut by the rise of Massachusetts Sen. Elizabeth Warren and watched as both Ms. Warren and Vermont Sen. Bernie Sanders hauled in significantly more money, fueled by small-dollar donations...