Matthew Taylor, a Conservative county councillor who finished second in the PCC election, said he was "shocked" to learn of the former detective's plans for a long run in theatres.
"If there is a major incident, you would expect the police and crime commissioner to be there, front and centre," he said.
"Not operationally, that's down to the chief constable, but in terms of lending his support, communicating with the public, being active and visible. "
The BBC has asked Sutton for comment.
A spokeswoman for the Office of the Police and Crime Commissioner for Norfolk said his activities were "being undertaken in a private capacity...in his own time" and that he would "remain available and contactable should any business require his attention".
"Based on these assurances, we are satisfied that appropriate arrangements are in place and that there is no adverse impact on the performance of the commissioner's official responsibilities," she added.
Next week Sutton is due to launch a public consultation on policing priorities for Norfolk.
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