A 2013 Statistics NZ census of disability found 380,000 people with impaired hearing.
The deaf groups asked the Human Rights Commission to investigate the issue today. They have also called on Broadcasting Minister Amy Adams to "address this matter with extreme urgency".
Sky TV government relations manager Chris Major told the groups on Monday that it had proved impossible to build a connection with the agency that provides captioning for TVNZ, Able, in time for the cup.
"Captions are not being provided by the host broadcaster (ITV) on the global RWC feed, so would need to be created in NZ," Ms Major told Mrs Carroll.
"There is currently no facility between Able and Sky to create a live captioned feed (as you know, Able is currently only funded to do live captioning for one broadcaster - TVNZ).
"With the time and resources available, that facility cannot be established in time for RWC. It requires establishing a secure, point-to-point VDSL connection between Sky and Able. Able would also need to invest in additional infrastructure to deliver live captioning to Sky (which they estimate to be in the vicinity of $100,000).
"Sky does not currently have the technology required to embed captions in the Sky Sport channels and further capex would be required to enable this."
However she said Sky was "committed to providing captioning where it is possible, and to steadily increasing the amount of captioned content on the Sky platform.".
She said the company had agreed to provide captions on two new channels, TLC and Discovery Turbo, and was "actively working with Able to get captions on some of our Prime Television content, particularly the local programmes on Prime that receive NZ on Air funding."
Mrs Carroll confirmed that captioning was currently provided mainly to TVNZ at a cost to the Government of $1.5 million a year.
"We are only talking $100,000 here," she said. "We are not talking a lot of money if you look at $26 million for a flag, or if you look at Sky TV's profit."
Broadcasting Minister Amy Adams said the level of captioning for free-to-air programmes had more than tripled over the last decade.
"NZ on Air funds Able to the tune of $2.8 million per year to caption TV and audio programmes and currently 100 per cent of prime time TV programmes are captioned," she said.
"However I understand the technology required to caption live television (including sports events) is not readily available in New Zealand.
"At present delayed games shown on Prime cannot be captioned because Prime does not have captioning equipment.
"I encourage broadcasters and other providers of content to continue to their work to overcome these technical difficulties so we can improve accessibility for deaf and hard of hearing people."