The disruption of oil supplies was caused by the closure for maintenance of the Khartoum plant - the only refinery in Sudan producing JetA1 fuel.
The WFP said aid workers in the town of Wau had reported an increase in severe cases of malnutrition as a result.
"This could not have happened at a worse time for the people of Sudan," said WFP Country Director Ramiro Lopes da Silva.
"This is a tragedy for hundreds of thousands of people," he said.
The WFP said donors had provided only 60% of the funds it requested to carry out its feeding programme in the south, following the peace deal signed earlier this year to end decades of civil war.
The agency said "sporadic looting was another problem".
It also said aid operations for more than two million people in Sudan's western region of Darfur were being affected by the shortage of jet fuel.