What, exactly, was the professional role of The Summoner?

He was church official. He was related to a branch of the church. The Summoner actually blackmailed people for profit.

The job of a summoner, to which the Friar objects, is to issue summons from the church against sinners who, under penalty of excommunication, pay indulgences for their sins to the church, a sum which illicit summoners often pocket.

http://www.gradesaver.com/the-canterbury-tales/study-guide/section8/