Pat Studstill and Tom Yewcic are tied for the most total touchdowns by a punter in a game, with 2 touchdowns.

who has the record for to touchdown scored in a game of punters
Interpreted as:
Who has the record for to touchdown scored in a regular-season game of punters?
12/20/1970Studstill11/30/1962Yewcic11/11/2012Koch9/13/2009Smith12/12/2004Larson11/16/2003Smith11/14/1993Kidd9/10/1989Prokop10/13/1986Hayes12/21/1980McInally11/30/1980McInally12/18/1977McInally12/10/1977McInally12/4/1977McInally11/28/1976Seiple9/22/1974Parsons10/14/1973Van Heusen9/17/1972Van Heusen12/20/1970Latourette10/19/1969Bradley10/20/1968Studstill9/29/1968Latourette2211111111111111111111
Results may be incomplete. StatMuse has complete game-level data for touchdowns going back to the 1950 season.

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