Former election agent Alison Hernandez is facing questions over her role in submitting expenses for Tory MP Kevin Foster in 2015.
Plymouth City Council Labour leader Tudor Evans said her decision to proceed with a swearing-in ceremony on Tuesday was "arrogant".
Ms Hernandez defended taking the oath.
The Conservative party has admitted failing to declare a total of Â£38,000 of general election expenses for accommodation costs during its 2015 election campaign, blaming an "administrative error".
On Friday, it emerged eight police forces were investigating whether Conservative MPs filed election expenses illegally.
Devon and Cornwall police is bringing in another force to investigate the claims against Ms Hernandez.
Mr Evans said an "urgent" meeting was necessary to discuss "the many issues that have arisen".
He said: "Police and crime commissioners are supposed to keep an eye on the police on behalf of the public.
"If a member of the constabulary was in the same situation as this police and crime commissioner, they would already be suspended."
Ms Hernandez claimed the seat as PCC for Devon and Cornwall with 91,036 votes.
Responding to the concerns, she said: "I've had over 91,000 people elect me to office. I'm here for the people of Devon and Cornwall and the Isles of Scilly.
"I'm here to do a job - the police need support."
Mr Hernandez, who has previously faced calls to "stand aside", said she would review station closures, help victims of crime and increase police visibility in her new role.

The news continued the club's dismal week after Tuesday's confirmation that their 12-point Premiership deduction would remain following a failed appeal.
A day later, Portadown conceded two late goals to lose 4-3 against Crusaders in the League Cup.
"I believe the time is right to step away - I've done all I can to move the club forward," said McGibbon.
After serving an interim boss following long-serving Ronnie McFall's resignation in early March, former Manchester United defender McGibbon was appointed manager in late April.
However, six weeks later Portadown were handed a 12-point deduction for this season's Premiership after an Irish FA disciplinary committee ruled that they had paid player Peter McMahon even though he had been registered on amateur forms.
Portadown were already facing a transfer embargo in addition to being fined £5,000 because of a further registration breach involving Gary Twigg.
The Shamrock Park club were handed a total ban from all football related activity in late July for a failure to pay the £5,000 fine and their first scheduled league game of the season against Coleraine on 6 August was postponed.
Portadown appealed against both penalties but their pleas were rejected by an Irish FA appeal panel last Tuesday.
"As a former player both at youth team and first team level, I am devastated to see the current state of the club," added McGibbon.
"Recent events have had a demoralising effect on players, management and supporters alike.
"Despite player restrictions and ongoing administrative issues, I have been working hard with Vinny, the players and the rest of the management staff to improve fortunes on the pitch.
"However, I have found that my roles and responsibilities have been taken up more as a general manager than from a first team management perspective.
"Vinny and the rest of the backroom team have been terrific in helping with training whilst I have prioritised and dealt with other issues."
The new Devon and Cornwall police and crime commissioner has faced fresh criticism for taking the role while subject to an expenses investigation.
[NEXT_CONCEPT]
Crisis club Portadown are looking for a new manager after Pat McGibbon resigned on Friday evening.