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".
A war memorial in Devon has been burnt in a "sickening act of mindless vandalism", police say.