ITHACA, N.Y.—With the odds quickly growing againstSen. Bernie Sanders’s bid for the Democratic presidential nomination, some in his army of online advocates are vowing to continue the fight even if he drops out, attacking Joe Biden and threatening to back a third-party candidate.

Victor Tiffany helps lead that loosely connected digital militia from an aging desktop computer in his upstairs home office here. The 64-year-old IT technician and protest-movement veteran runs a “Bernie or Bust” network that has been a potent force...