Two care workers tell ITV News Correspondent Rachel Younger they were stuck in a queue at a petrol station for 40 minutes in the middle of their shift
The president of the AA has said some petrol stations are "rationing" the amount of fuel customers can buy and prioritising essential workers looking to fill up.
Some garages are experiencing a shortage of supply caused by issues with transporting fuel from distribution terminals to forecourts - things have since worsened due to panic buying.
The issue stems from a lack of HGV drivers - something the government is hoping to tackle by relaxing visa rules to allow 5,000 more foreign drivers into the country until Christmas Eve.
On Saturday there were further scenes of queuing vehicles, despite official messages not to panic buy, prompting garages to take matters into their own hands according to AA boss Edmund King.
'Some garages are giving priority to essential workers' says AA president
Mr King told ITV news: "Some petrol stations are rationing the amount of fuel and only allowing £30 to be purchased and that means it does go further so that essential workers can get fuel."
He added: "Some garages are giving priority to essential workers, some delivery vehicles have been given priority."
The motoring boss stressed, however, "there is plenty of fuel at source" and so issues would be a "short-term thing".
Mr King said if garages are closed on Saturday they should be reopened "within 24 to 48 hours once they are refilled".
ITV News' UK Editor Paul Brand reported that a Kent-based care company providing end of life said their workers were unable to get to patients.
"Our patients are dying and if this continues their pain management will suffer," an individual from the company told him.
'We've got a problem on the ground and we need to solve it... people are dying and we need to get them help' - Jane Townson, chief executive of the Homecare Association
Emergency services on Saturday urged drivers to be "sensible" about refuelling and only do so if they need to.
Lincolnshire Police said long queues were "a potential hazard for vehicles making other journeys, and may cause difficulties for emergency services trying to reach people in need."
North Wales Police asked people not to panic buy and asked drivers not to "form queues and cause unnecessary obstructions in roads".
Will temporary visas for foreign HGV drivers solve the problem? - ITV News Correspondent Rebecca Barry
South Central Ambulance service said its crews have had to join the queues with others to refuel - which in some cases has taken some time.
Essex Police said it had received more than 100 calls in the past 24 hours from people stuck in traffic, the force urged the public to only call in emergencies.
ITV News first revealed on Thursday that BP's Head of UK Retail, Hanna Hofer told a meeting organised by the Cabinet Office the company was preparing to restrict deliveries of fuel.
She said it was important the government understood the “urgency of the situation” which she described as “bad, very bad”.
What action is the government taking?
The government has announced a temporary visa scheme will be brought in to help fix the lorry driver shortage, as ministers hope to rescue Christmas from supply shortages.
A total of 5,000 HGV drivers and 5,500 poultry workers will be able to take up employment in the UK until Christmas Eve, in a bid to keep supermarket shelves stocked with turkeys and toys and counter delivery difficulties at petrol stations.
Transport Secretary Grant Shapps said the changes, with the visas available from next month, would “ensure preparations remain on track” for the festive season.
Mr Shapps said: “This package of measures builds on the important work we have already done to ease this global crisis in the UK, and this Government continues to do everything we can to help the haulage and food industries contend with the HGV driver shortage.
“We are acting now but the industries must also play their part, with working conditions continuing to improve and the deserved salary increases continuing to be maintained in order for companies to retain new drivers.
“After a very difficult 18 months, I know how important this Christmas is for all of us and that’s why we’re taking these steps at the earliest opportunity to ensure preparations remain on track.”
Labour has said it would back the introduction of short term visas for foreign lorry drivers but said the government has already been too slow to react.
On Friday BP said that around 20 of its 1,200 petrol forecourts were closed due to a lack of available fuel, with between 50 and 100 sites affected by the loss of at least one grade of fuel.
A “small number” of Tesco refilling stations have also been impacted, said Esso owner ExxonMobil on Friday, which runs the sites.
The EG Group, which has 341 petrol stations in the UK, confirmed on Friday it was imposing a £30 limit on customers “due to the current unprecedented customer demand for fuel”.