Vorderman, who said she was told to take a 90% pay cut to stay on Countdown and was given 48 hours to decide, leaves the cult gameshow at the end of this year.
Yesterday it was announced that the broadcaster is due to mark the departure of hosts Vorderman and Des O'Connor, who is also leaving, with a special show.
The farewell is being seen as part of a move by Channel 4 to repair its relationship with Cambridge-graduate Vorderman, 47, who has been with the show for 26 years.
The broadcaster said it was looking for applicants with "charm and charisma to appeal to viewers" and "excellent numeric and literacy skills".
Previous TV experience is not necessary and applicants have been asked to apply and submit a photograph of themselves through the show's website.
Short-listed "maths whizzes" will attend auditions with Countdown producers and the new recruit will appear on screen in January next year.
Channel 4 head of daytime Helen Warner said: "We realise that Carol's shoes are extremely difficult ones to fill, as she has done such an amazing job on Countdown over the past 25 years.
"However, just as we found Carol through an ad in a local paper, we really hope we can do the same again and find another outstanding mathematical mastermind.
"We know many Countdown viewers have great heads for numbers, so we are looking forward to seeing how talented they really are."
Countdown was the first programme seen on Channel 4 and nearly 5,000 episodes of the show, a mixture of word games and mathematical problems, have been filmed.