The petition, which calls for the UK to share the burden with Europe, was launched on parliament's official petition website. It gained almost 70,000 backers overnight after a picture of a dead 3-year-old child washing up on a Turkish beach provoked outrage on Thursday.
Any British citizen or UK resident can start a petition on the government's website and the government will respond to any that receive more than 10,000 signatures. Since the petition has now received more than 100,000 signatures, the motion will have be considered for debate in parliament.
Labour leadership candidate Andy Burnham called for a debate to be held in parliament next week about accepting more refugees.
Katie Whyte told BuzzFeed News that she launched the petition after she felt "helpless" when she saw images of refugees fleeing conflict zones and trying to get to Europe.
Whyte, 36, said the government's response had been wrong and that it should welcome more refugees into the country.
"I don't know how we can turn our backs on the situation," she said. "It's completely unacceptable."
She added: "Ultimately I'm just disgusted with the government and I can't believe the fact that we celebrate that we liberated concentration camps but it's like a modern day apathetic equivalent.
"There's no dictator but there are people who are malnourished or at risk of disease and it seems completely inhumane to me for us to just put up fences and ignore the problem. There's room in our country."
Hundreds of people have been trying to enter the UK through Calais, and many newspapers have portrayed them in a negative light. In response, the government said it would work more closely with France and put more resources towards security at the border.
But Whyte's main objective is to shift opinion. "I don't know how effective a petition is," she said, "but if we can get enough people saying that this [not welcoming those fleeing conflicts] is not British, then maybe there can be some positive news instead of this anti-asylum opinion."
The UK government has come under sustained criticism from European countries in recent weeks for its failure to take in more asylum-seekers and refugees who are fleeing conflicts.
Britain has only accepted 216 Syrian refugees into the country as part of the Vulnerable Persons Relocation scheme. The Washington Post pointed out that if they were all brought together, they could easily find space on a London underground train.
When launching the petition, Whyte said she was hopeful that Britons would empathetic and that others would help if they were struggling and forced to flee to another country. "It could be us in this situation and I know if this was the case that we would hope that people would help," she said.
More than 100,000 people entered Europe last month alone, according to EU figures, many of whom were fleeing conflicts in Syria and Afghanistan.
In a speech on Tuesday, Labour leadership candidate and shadow home secretary Yvette Cooper said Britain should accept 10,000 more vulnerable refugees into the UK.
"We seem paralysed to respond," she said. "Stuck in the troubled politics of immigration when this is about asylum instead."