Joshua Spencer was sentenced to nine weeks behind bars on February 7 for sending a message in which he claimed he was paying "crackheads" £100 to hurt Ms Cooper, Labour's MP for Normanton, Pontefract and Castleford.
Councillor Nadeem Ahmed, who leads the Tory opposition on Wakefield Council, has branded the incident "disgusting" and said Spencer should have been suspended by the party "much quicker".
It was revealed in court that Spencer was allowed to attend December's General Election count in Wakefield, where Ms Cooper was present.
Asked about the incident this week, Coun Ahmed said: "My personal view is that any form of bullying and harassment in politics, whether it's against Yvette Cooper or anyone else, is completely unacceptable.
"On behalf of the Wakefield Conservative group I'd like to apologise to Yvette and her staff for what happened, and I'd like to meet them to make that apology personally if I can.
"I think we do have to look at our procedures in terms of suspensions, because if it had been one of my councillors accused of that they would have been suspended straight away.
"I have no personal issue with the Labour party in Wakefield and I've always found Yvette's staff to be open and approachable when I've met them at events.
"I might not agree with her views on socialism or the EU, but political argument should never be brought to that level of abuse."
Coun Ahmed said he was keen to sit down with Ms Cooper to discuss making political discourse less toxic, amid high-profile incidents of abuse up and down the country in recent years.
Coun Ahmed also addressed criticism of Conservative MP Andrea Jenkyns, who gave Spencer, from Knottingley, a character reference ahead of his case.
It was revealed that Spencer, who has now been expelled by the party, had had serious mental health problems and had struggled since his father's suicide in 2015.
Although he did not directly condemn Ms Jenkyns for giving a reference, Coun Ahmed said: "Now and then I get asked to write character references, and in most cases I refuse, often because I don't know that person very well.
"The only one I have given was for somebody I'd known since I was 13 years old.
"I think Andrea's reference was around the fact he (Spencer) had suffered from mental health issues, so I understand why she gave it."
In response, Ms Cooper said: "I welcome the apology from the leader of Wakefield Conservatives and the apology from the Home Secretary in Parliament earlier this month.
"Violent threats should have no place in public life - and this should be something all of us can agree on whatever our party politics.
"I have expressed concern about how long it took the national Conservative Party to deal with this. All parties need to do more to tackle threats and abuse in public life.”
Local Democracy Reporting Service