Prolife groups often hold demonstrations outside buildings to show their objection to terminations but deny harassing anyone.
The petition wants 'buffer zones' created outside clinics meaning protestors would have to stay a certain distance away from the building.
It says: "Abortion is not illegal in the UK and nor should it be. Women should be able to visit these clinics without fear, harassment or intimidation.
"It can be hard to go through those doors, let alone if you have to get past protestors.
"It is up to the pregnant woman to decide what she does about her pregnancy. Those decisions need to be made in private and not with a group of protesters filming and/or hounding her."
More than five million people have watched a YouTube clip showing a pregnant woman challenging protestors outside the clinic.
Often groups who protest will use large graphic posters showing the aftermath of abortion.
Pro-life group Abort67 claims a change in the law to stop protests would be a bad thing.
Spokesman Andy Stephenson said: "The ability to speak to an empty room or speak to people who your message isn't aimed at - that's not free speech.
"It's a direct attack on free speech disguised as a compassionate move for women."