BERLIN, N.H.—New Hampshire voters take their first-in-the-nation primary very seriously, ribbing their early-voting compatriots in the Midwest with the expression, “In Iowa, they pick corn. Here, we pick presidents.”

But now, the success of Donald Trump, who has topped polls of Republicans here for nearly five straight months, is unnerving those who cherish the small state’s tradition of shoe-leather campaigns, with candidates trekking from bingo hall to barnyard to answer policy questions from voters face to face.

The...