Boris Johnson is being urged to give priority to petrol to NHS staff and police officers, amid warnings that pump chaos could put lives at risk.
Unions are calling on the PM to set aside fuel stocks for emergency services workers so that they can continue with their work.
The British Medical Association said there was a growing risk that some health workers would not be able to reach “very sick patients at home”.
The petition comes after an ambulance driver was abused by motorists after taking two days to fill petrol.
Meanwhile, at a service station in Hackney, east London, police were forced to jump into a large queue to avoid running out of gas.
Understands that Ministers are not currently considering providing emergency workers with exclusive access to fuel.
Today a police force was forced to beg people to stop endangering others at about 999 in the queues of petrol pumps.
Essex Constabulary was inundated with more than 100 calls to its emergency hotline in less than 24 hours.
Chief Superintendent Jenny Barnett warned: “If our call handlers are dealing with calls about traffic build up, they are unable to take calls from people who really need our help.”
Number 10 today again asserted that there is “no shortage” of fuel in the country and urged people to stop shopping unnecessarily in panic.
Read our Petrol Crisis Live Blog for live updates on the crisis
A retired fire commander also warned motorists that it could be dangerous to fill their shoes with jerry cans.
Motor store Halfords revealed that sales of such containers jumped 1,656% this weekend, as Brits stockpiled fuel.
BMA boss Dr Chand Nagpaul said the crisis was becoming so dire that “there is a real risk that NHS staff will not be able to do their jobs”.
He said: “Emergency and essential workers depend on fuel both to go to work and for their jobs.
“Whether it is for hospitals, practices and other health care settings, or for ambulances to reach people in urgent need of care and GPs to visit very sick patients at home.
“Everyone will have their reasons for needing to fill up, but as pumps run dry, there is a real risk that NHS staff will not be able to do their jobs and provide vital services and care to those who need it urgently. Is.
“While the government has said it plans to address the shortage of HGV drivers to transport the fuel, the consequences will not be immediate.
“Health care and essential workers must be prioritized to fuel up so they can continue their vital work and guarantee care to patients.”
Unison, one of the UK’s largest unions, said the government should now “designate fuel stations for the sole use of key workers” like nurses.
Secretary-General Christina McNea said: “The government has to take control.
“It is not a good minister who is wasting time in a pointless blame game or pretending that there is no problem.
“Essential workers need to be able to get their jobs done so that they can continue to provide the services that many people rely on.
“Ambulance crew, nurses, care workers, teaching assistants, police staff and other key staff should not be stranded or queue for hours just to reach a pump.
“The government can now address this problem by using emergency powers to designate fuel stations for the sole use of key workers.”
London Mayor Sadiq Khan echoed those calls.
He added: “As the current shortage in fuel delivery affects petrol stations across the capital, it is essential that key workers be able to obtain fuel to go to work and provide the services our city needs.
“In the fuel crisis of September 2000, the government brought in rules designating specific filling stations for essential workers, to keep the capital running.
“The government should immediately consider taking necessary steps to take such measures, so that key workers who have to go to work can do so.”
An “emotional” NHS nurse also joined the call for healthcare workers to keep supplies separate.
Paul Mulligan, who works as a nurse at the Rochdale Infirmary, feared that coworkers might not be able to go to work any time soon.
Speaking from the queue at a Tesco petrol station in Oldham, the 54-year-old said: “I am very upset. People are tense.
“There should be dedicated supplies for ambulances for NHS staff”
And set the trucks on fire because it’s going to get worse.”
Ministers and medical experts today reassured Britons that there is no need to panic buying petrol.
The Royal Pharmaceutical Society said the delivery of medicines for people’s prescriptions is going on as normal.
Its chairman, Professor Claire Anderson, said: “People should order and collect their prescriptions in the usual way.
“We are not aware that problems with the fuel supply are preventing patients from taking their medications.
“If you have any concerns please speak to your local pharmacist and their team, who will be able to help and reassure you.”
Environment Secretary George Eustice said that as soon as people stop panic buying, the petrol crisis will end.
He said: “We have a lot of petrol. Production at the refineries is at normal levels, and we have a lot of petrol in the shops.
“The most important thing is that people buy petrol in general. There is no shortage.
“The only reason we don’t have petrol in the forecourt is because people are buying petrol when they don’t need it.”
He also did not pay heed to the reports which said that the PM is going to bring the army to deliver fuel to the service stations.
The crisis has been sparked by a shortage of HGV drivers.
Amidst the chaotic scenes, there has also been a brawl between the warring drivers over the pumps.
Number 10 is pinning its hopes on the fact that most people who are willing to panic buy will have already stocked up.
The PM’s spokesperson said: “Should the public return to their regular practice of refueling which will contribute significantly in ending the challenges we face.
“We fully understand the public’s concern over this and will continue to do everything possible to assure them that there is no fuel shortage.”
But industry owners fear the fuel crisis could last at least another week and warn that all service stations outside the motorway are now dry.