No, as a PP, he cannot accept pay for flying a 91.321 flight. He can't even be the pilot without pay. The aircraft operator can be paid for the transportation, but a Private Pilot cannot be PIC. That section provides relief only from the requirements of Parts 121, 125, and 135, not the Part 61 requirement that the PIC of an aircraft carrying persons for hire/compensation hold a CP or better. So, he can provide the plane, and be paid as required by Federal election laws by the candidate for providing the transportation, but must get a CP or better to be the pilot.
BTW, this section was added primarily to allow corporations to carry candidates in their bizjets without running afoul of Federal campaign contribution laws. In those cases, the aircraft are flown by professional CP/ATP pilots in the corporation's employ. As an unintended consequence, it allows privately owned planes to do the same thing, but only with CP or better pilots. It was not intended to bypass the 61.113 restrictions on PP's flying airplanes with paying passengers, and does not do so.