This could include fining motorists who stop on a box junction or those who drive in bus lanes.
Currently only police officers can punish motorists if a vehicle is moving.
The Welsh government won the right to grant councils those extra powers last March and Cardiff is now making a bid.
Local authority traffic wardens are currently only able to issue penalty charge notices (PCNs) for parking offences.
Cardiff council's cabinet has given the authority the green light to make a request for the additional powers, which it says would improve traffic flow and ease congestion.
It is expected that income generated from the scheme would help pay for the service and the staff who run it.
A Cardiff council spokesman told the BBC's Welsh language news website Newyddion Ar-lein: "The intention of all of these measures is to ensure drivers drive responsibly and in accordance with the basic principles of the highway code.
"If the Welsh government approves, specific emphasis will be placed on enforcement of bus lanes to ensure that public transport runs efficiently and is an attractive option for commuters.
"Enforcement of yellow box junctions is also necessary to avoid grid lock which disrupts traffic flow and to ease the frustration caused to the majority of drivers who abide by the law."
A Welsh government spokesman said it had yet to receive the council's request, but was aware of it.
The process of considering the request could take up to four months, the spokesman added.
Many councils in London use traffic cameras to enforce the rules.
One camera on a yellow box junction in Bagley's Lane in Fulham caught 29,000 drivers last year making over Â£2m in fines for Hammersmith and Fulham council.
Cardiff council said it was unsure if it would ask for powers to use cameras to enforce the rules.
If granted, the authority could start using the new powers towards the end of the summer.

Wayne Esmonde, 35, had his face put on South Wales Police's Facebook appeal over an alleged assault.
But the picture shows shaven-headed Mr Esmonde posing for the camera looking wide-eyed and staring intently.
Mr Esmonde, who asked for officers to take the post down, handed himself in at Swansea Central police station on Thursday.
He had written on their Facebook page: "I am him. Not a very flattering mugshot.
"I'd appreciate it if you'd take this post down. Innocent until proven guilty and all that.
"I've spoken to my solicitor and they will advise me what to do next. Thanks."
Mr Esmonde, of Fforestfach, Swansea, was wanted in connection with an assault on 18 July and South Wales Police have since updated their Facebook post.
Traffic wardens in Cardiff could become the first outside London to get powers to issue tickets for moving traffic offences.
[NEXT_CONCEPT]
A man who asked police to take his mugshot off their wanted appeal because it was unflattering has been arrested.