Wreaths placed at the memorial in Crediton, Devon, including one for L/Cpl Jamie Fleming, 21, were left scorched by the fire.
Two teenagers - a boy and a girl - were seen running away from the scene towards a park early on Monday morning.
L/Cpl Fleming's father said he "could not come to terms" with the attack.
Among the wreaths was a family tribute to the soldier, who died in a motorcycle crash earlier this year after serving in Afghanistan.
His father Stewart, from Crediton, told BBC News: "As a family we were very angry at the wanton vandalism.
"We are still raw over Jamie's death.
"I am lost for words."
Insp Jane Alford Mole, from Devon and Cornwall Police, said: "The wreaths suffered burn damage and there are a number of scorch marks on the war memorial.
"This is a sickening act of mindless vandalism and we are desperate to identify those responsible."
The two suspects are described as aged about 14 to 15 years old and were both wearing hooded tops.
Local resident Gillian Ponsford, whose two young grandsons laid a wreath at the memorial which bears the names of their great, great uncles, said she was "horrified".
She said that wreath had been one of those burnt and described the vandalism as "unbelievable" and "a sad day for Crediton".

Colin Ferrie, 52, of Waters Walk, Apperley Bridge, Bradford has also been charged with possessing a class C drug.
He will appear at Leeds Magistrates' Court on 18 August alongside Scott Pendlebury, of Daisy Fields, Bradford, who is charged with the same offences.
It follows claims made by the Mail on Sunday newspaper last year about Dr Ferrie's alleged drug taking.
West Yorkshire Police said a 23-year-old man, from Bradford, who was also arrested as part of the investigation, had been released from bail and would face no further action.
Another man who was interviewed under caution after voluntarily attending at a police station will also face no further action.
Leeds Teaching Hospitals NHS Trust said Dr Ferrie, a consultant paediatric neurologist specialising in epilepsy, no longer worked there.
A war memorial in Devon has been burnt in a "sickening act of mindless vandalism", police say.
[NEXT_CONCEPT]
A former children's doctor from Leeds General Infirmary has been charged with possession of cocaine.