The COP26 conference, which was put back from last year due to the Covid crisis, is scheduled to take place from 31 October to 12 November in Glasgow, Scotland.
"Our concern is that those countries most deeply affected by the climate crisis and those countries suffering from the lack of support by rich nations in providing vaccines will be left out of the talks," CAN executive director Tasneem Essop said.
Farhan Haq, a spokesperson for UN secretary-general Antonio Guterres, said the scientifically established urgency of combatting climate change meant a further delay of COP26 "is no longer feasible."
"The global scientific community has made clear that climate change is now a global emergency and only an urgent and major step up in climate action can keep the goals of the Paris Agreement within reach and protect the most vulnerable countries and communities from worsening climate impacts," Haq said.
COP26 host Britain said in June it would offer vaccines to delegates who need them and has since said vaccinations under this programme would start this week.
However, CAN in a statement said that Britain has been slow in delivering vaccines and many countries are likely to miss out as a result.
The issue is made worse, it said, by the need for unvaccinated delegates to quarantine in a hotel for 10 days if they are arriving from what Britain has identified as so-called red list countries.
Britain's COP26 president, Alok Sharma, said Britain would pay the quarantine costs and it was vital talks go ahead as planned.
"Ensuring that the voices of those most affected by climate change are heard is a priority ... if we are to deliver for our planet, we need all countries and civil society to bring their ideas and ambition to Glasgow," Sharma said in a statement