BERLIN, Vt.—Aidan King took a break at the Fresh Tracks Farm Vineyard and Winery from picking St. Croix grapes, found a spot on the mountain with good cell service, wiped the purple juice off his hands, and turned his attention to Bernie Sanders.

Mr. King, 23, is the co-founder of one of the largest online organizations supporting the Democratic presidential candidate, a group that helped the Vermont senator raise more than $26 million in the third quarter fundraising period from 1.3 million donations.

The...