In the email, Mr Campbell said his comments were "a poor lapse of judgment".
Ms Mathers accepted his apology and said she was not offended.
"But these kind of statements do reflect people's assumptions about deaf and disabled people and create major barriers for us in participating in our own democracy.
"This is one of the reasons it is so important to have people like myself in Parliament."
The mother-of-three, 45, was born profoundly deaf after her oxygen supply was cut during a difficult birth.
"I aim to be an advocate for deaf and disabled people as well as a strong voice for the environment. Just by participating in parliamentary business, I will hopefully provide inspiration for young people with disabilities of what is possible to achieve."
She hopes her participation in the debating chamber will be helped with captions on Parliament TV, which would also open up Parliament to thousands in the deaf community.
"I am also really keen to see much greater use made of sign language in Parliament. It is our third official language and the deaf have a right to access their Parliament in their language."
A mathematics and conservation ecology graduate, Ms Mathers joined the Green Party in 2002 during her opposition to the Central Plains water scheme in Canterbury, which she said would have caused a huge increase in intensive dairy farming.
She hopes to be responsible for disability issues, though the Greens have yet to officially allocate responsibilities among its caucus, which has gone from nine to 14 MPs.
Ms Mathers comes into Parliament at the expense of National list candidate Aaron Gilmore.
