The appeal, which says "fireworks cause alarm, distress and anxiety to many people and animals", asks that only professional displays to be allowed to continue.
More than 2,900 people had signed the parliament.uk petition by Monday afternoon. It must garner 10,000 signatures before the government must respond.
It calls on the government "to make appropriate provision to secure that the risk of public use is the minimum that is compatible with fireworks being used."
Organiser Julie Doorne told The Independent: "We're asking the government to look again at fireworks legislation. We have never said we don't want [official] displays, but many come out of the blue."
She said random displays throughout winter caused distress to people, not just animals.
"Some people are really suffering from fireworks all the time. If you're deaf, you wear a hearing aid and a firework goes off next to your house, it can hurt," she said.
"Many deaf people remove their hearing aids on bonfire night," she added.
"One lady contacted us to say her husband took his hearing aid out from October to January."
Some autistic children struggle with loud noises and fireworks can make them stressed, she said, adding that parents found it difficult to explain to them why the displays were happening throughout the winter months and not just on bonfire night and New Year's Eve.
It is not the first time campaigners have called for an end to members of the public handling fireworks.
A similar petition launched in 2016, which argued that thousands of people were "terrorised" by unofficial fireworks displays, gained 161,160 signatures.
Following the 2016 petition, Conservative MP Philip Hollobone, told a parliamentary debate all displays should be professionally licensed in future, telling the Commons that amateur back garden fireworks displays were “rubbish."
But the Government rejected the plan, claiming amateur fireworks users would not be “deterred by further regulation” and that restrictions could lead to job losses and a drop in “legitimate sales”.
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