Hamas issued a statement condemning the attack as "a horrific massacre" and accused Israel of deliberately targeting personnel tasked with protecting humanitarian aid.
BBC News has contacted the Israel Defense Forces (IDF) for comment.
A small amount of food has been allowed to cross into Gaza this week: around 130 lorries carrying aid have crossed the border in the last three days, after an 11-week blockade was partly lifted by the IDF.
The UN says 500 to 600 trucks of supplies a day are needed in Gaza.
LIVE: Gazans tell BBC more aid needed as Israel eases blockade
International agencies, including the UN and the WFP, have repeatedly warned that the growing insecurity is hampering the delivery of desperately needed food and medical supplies to the population - the majority of whom are displaced.
Israel says the blockade was intended to put pressure on Hamas to release the hostages still held in Gaza. Israel has also accused Hamas of stealing supplies, which the group has denied.
The WFP said 15 of its aid trucks were looted overnight on Thursday, and that "hunger, desperation and anxiety over whether more food aid is coming is contributing to rising insecurity". The organisation called on Israel to help ensure the safe passage of supplies.
Philippe Lazzarini, the head of UNRWA, an agency that supports Palestinian refugees, wrote on X that no one should be "surprised let alone shocked" that aid had been looted because the "people of Gaza have been starved [and] deprived of the basics including water and medicines for more than 11 weeks".
Earlier on Thursday, angry and hungry Palestinians crowded outside bakeries in Gaza in a desperate attempt to obtain bread, but the situation quickly descended into chaos, forcing distribution to halt.
It forced most bakeries to suspend operations, citing a lack of security.
Many residents across Gaza voiced growing frustration over the aid distribution method and criticised the WFP, which oversees food deliveries.
Some called for an immediate shift from distributing baked bread to handing out flour directly at a rate of one sack per family.
Locals argue that distributing flour would allow families to bake at home or in tents - which, they say, would be safer than waiting at the overcrowded aid centres.
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