The deaths occurred in a displacement camp in the northern Warrap state.
It comes after the UN's World Food Programme (WFP) said it had suspended rationing programmes in parts of the country due to a lack of funding.
In June, the body said it needed $426m (£352m) to continue distributing food.
But that aid has not been forthcoming as the international community has focussed on other overlapping crises, including the Russian invasion of Ukraine.
Sami Al Subaihi, a Médecins Sans Frontières worker at the camp, said that one of the children was just five years old.
"I find the mother of one of the children sitting by her five-year-old son's small, freshly dug grave," he said. "Her three other children, all very thin and weak, sit at the entrance of the family's improvised shelter."
He added that more than 20,000 people living at displacement camps were at risk of starvation.
"In one camp I see people collapsing, physically exhausted. They clearly haven't had enough food for a while. I don't see anyone cooking or any food stored in any of the shelters."
A WFP official told the BBC the body has "suspended assistance to 1.7 million people due to being underfunded, but we continue [to help] 4.5 million people".
They said the suspension was spread out across different regions of the country and added that more than 60% of the population of 11 million now faced "food insecurity".
On 14 June, the body said that it had "exhausted all options before suspending food assistance, including halving rations in 2021, leaving families in need with less food to eat".
South Sudan has been wracked with violence since it achieved independence in 2011 and this has contributed to food insecurity, despite the formation of a unity government in 2020.
The situation has also been worsened by four consecutive seasons of flooding that have destroyed homes and farmlands, and displaced tens of thousands.