The vans were destroyed in the attack outside an Iceland store in Arnside Road, Southmead, Bristol, on Monday night.
Richard Walker, managing director of the retailer, said: "At a time when home delivery is literally a lifeline for some vulnerable people, this is sickening.”
In a statement, Iceland said the disturbances began at around 9.30pm.
"Police riot vans were called out and youths threw missiles at the police as they tried to disperse them," the retailer said.
"Later in the evening the two home delivery vans parked outside our Southmead store were set alight and the fire brigade attended to put out the fire.
"The vans will both have to be written off. We believe this to be an act of mindless vandalism rather than a targeted attack on Iceland.”
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It added: "While we can and will bring in replacement vans to maintain our home delivery service in Southmead, our entire van fleet is running at absolutely full capacity at the moment, and it is shocking and distressing that anyone would act so callously to put vehicles out of service at precisely the time when they are most needed to deliver food to the most vulnerable people in our society.”
Patrolling officers came across the Iceland van fire and called it in while officers were flagged down by a member of the public over the car damage in Strathearn Drive.
Chief Inspector Mark Runacres said: "Just hours after the prime minister announced that everyone in the country should stay home to save lives, it is beyond belief that anyone would be so reckless and thoughtless.
"Officers were already patrolling the area following recent instances of antisocial behaviour and we'll continue to do so.
"We put an order in place last night to give officers extra powers to disperse groups and will continue to use our existing powers to tackle this problem.
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"We are working with our communities and partners to identify and take appropriate action against those responsible. If you have any information which could help, get in touch.
"We believe that a group of teenagers were involved in these disturbances. I have one message for them, their parents and carers: stay at home. Save lives.”
Police are also investigating after an empty Asda delivery van was stolen from a depot in Weston-super-Mare and set on fire.