Although the trust would not disclose the amount it is seeking, local media reports suggested £300,000.
A spokeswoman at the charity's Bury St Edmunds office said it issued a writ to Korean Airlines in late June for damages in connection with two areas.
"We are recovering costs for the time when the forest had to close in 2000 and also for damage to conservation areas, including ancient trees," she said.
A national spokesman at the trust said there was no precedent for this type of action and that the trust considered litigation only in exceptional circumstances.
The flight was bound for Milan and Seoul, but crashed just minutes after take-off from Stansted Airport. Carrying depleted uranium within its tail structure, the plane narrowly missed crashing into local homes.
The explosion spread across a wide area of forest and countryside, which caused extensive damage and resulted in large clean-up costs.
A report by the Air Accident Investigation Branch found that the crash was caused by faulty equipment.