One reported that six garages in Ashford, Kent, were sold out, and long queues formed at others across the UK, from Edinburgh and the Wirral to Essex.
Petrol prices have risen to a record high of around £1.50 a litre of unleaded this week, after the tensions between Russia and Ukraine pushed up global oil prices.
Within hours of Russia invading its neighbour on Thursday, the RAC warned the cost was expected to surge past the £1.50 mark for the first time ever in the coming days.
But on Friday, some fuel stations started asking for up to 169.9p a litre.
Asda at Shoebury, Essex, reportedly ran out of fuel, after an estimated 100 cars were queued up at one point.
And the supermarket’s petrol station in North Hykeham, Lincoln, was also forced to close.
People who saw queues at forecourts slated the panic-buyers, accusing them of creating the very price rises they fear.
The UK buys only 6 per cent of its crude oil from Russia.
One HGV driver wrote on Twitter: “The petrol stations in Swindon have suffered panic buying again, so there’s no fuel now! Why? Putin doesn’t sell us oil. I may not be able to work Monday if this carries on, as my wagon needs diesel. Please everyone, calm down!”
Another driver wrote: “Supply and demand!! Please don’t go panic buying fuel bcos I really can’t afford it as it is.”
The Petrol Retailers’ Association, which represents independent retailers, said it would “continue to ensure the price of fuel was as competitive as possible”.
In September and October last year fears of petrol shortages became self-fulfilling, when queues forced many forecourts to close.
In Ukraine, too, petrol stations were swamped by drivers planning to flee with their families.
Some service stations rationed the amount of petrol each driver could buy.
Residents also reported panic-buying for food supplies and cash machines running out of money.