"It is an overwhelming response and demonstrates the strong public desire for MPs to tackle the subject," said Ms Street, who will today be confirmed as the new president of the Voluntary Euthanasia Society at its annual meeting in Wellington.
"The committee has undertaken to travel out of Wellington to hear submitters and that process is likely to begin in August."
Australian television and radio host Andrew Denton will address the society's AGM.
Mr Denton has been campaigning for voluntary euthanasia in Australia, including producing a 17-episode podcast about the right to die with dignity.
His interest came after he watched his father die slowly and in pain in 1997.
Mr Denton said research showed that there were the same issues in Australia and New Zealand.
"In your country, as in mine, the doctors are assisting people to die but without oversight and without guidance and without guidelines and scrutiny," Mr Denton said during an appearance on The Nation.
"In your country and in mine, elderly people are suiciding in desperate ways."
In response, Matthew Jensen, from the lobby group Care Alliance - an alliance including the Salvation Army - said Mr Denton was "spreading fear".
"What hospices and palliative care people do is they deal with the whole person, and they walk that journey with them, and that's what we should be doing, not giving them a lethal injection."
Legalising voluntary euthanasia would follow in the footsteps of a handful of overseas jurisdictions, including the Netherlands, Belgium, Luxembourg, Oregon and four other American states.
The issue was thrust back into the spotlight in New Zealand by Lecretia Seales. The Wellington lawyer, dying of brain cancer, asked the High Court to give her the legal right for a doctor to help end her life.
On June 5 last year, soon after being told that her court bid was unsuccessful, Seales died of her illness, at the age of 42. A judge ruled that only Parliament could make a law change allowing that legal right.
The following month, Seales' husband Matt Vickers was among supporters who presented a petition to Parliament, signed by Street and 8974 others, asking for its health select committee to fully investigate public attitudes to medically assisted dying in the event of a terminal illness or an irreversible condition which makes life unbearable.
Once it has finished considering the 22,000 submissions, the health committee can make recommendations, which are not binding, including that the Government consider legislation.
Euthanasia advocates are not confident of such a recommendation.
The best hope for a law change is if a member's bill put forward by Act leader David Seymour is drawn from the ballot.
He has expressed confidence that it would pass a first reading in a conscience vote.
