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 new Devon and Cornwall police and crime commissioner has faced fresh criticism for taking the role while subject to an expenses investigation.