The WFP said initial reports indicated that two people had died and several more were injured in the incident.
The UN agency appealed for an immediate scale-up of food aid "to reassure people that they will not starve."
Eyewitness video independently verified by Reuters shows large crowds of people pushing into the warehouse and removing bags and boxes as gunfire can be heard. It was not immediately clear how the people may have been killed or injured in the incident.
Under growing international pressure, Israel ended an 11-week long aid blockade on Gaza 10 days ago. It has allowed a limited amount of relief to be delivered via two avenues - the United Nations or the US-backed Gaza Humanitarian Foundation.
UN Middle East envoy Sigrid Kaag told the Security Council that the amount of aid Israel had so far allowed the UN to deliver was "comparable to a lifeboat after the ship has sunk" when everyone in Gaza was facing the risk of famine.
The United States has been trying to broker a ceasefire. Israel - which resumed its military operation in Gaza in March after a brief truce - continued strikes on Wednesday, killing at least 30 people, Palestinian health officials said.
"We are on the precipice of sending out a new term sheet that hopefully will be delivered later on today," US President Donald Trump's special envoy, Steve Witkoff, said. "The president is going to review it."
Separately, Israeli Prime Minister Benjamin Netanyahu said Mohammad Sinwar, Hamas' Gaza chief and the younger brother of the Palestinian militant group's deceased leader and mastermind of the October 2023 attack, Yahya Sinwar, had been killed.
Mohammad Sinwar had been the target of an Israeli strike on a hospital in southern Gaza earlier this month and Netanyahu said on May 21 that it was likely he was dead.
The Israeli leader announced Sinwar had been "eliminated" in an address to the Israeli parliament as he listed off names of other Hamas officials that Israel had killed over the past 20 months, including Sinwar's brother Yahya.
"In the last two days we have been in a dramatic turn towards a complete defeat of Hamas," he said, adding that Israel was also "taking control of food distribution", a reference to a new aid distribution system in Gaza managed by a US-backed group.
Netanyahu's announcement came as the Israeli military has intensified its war campaign in Gaza after breaking a fragile ceasefire with Hamas in March. Israel has said it aims to dismantle Hamas' governing and military capabilities and secure the release of hostages that are still held in Gaza.
The war erupted on October 7, 2023 when Hamas-led militants stormed out of Gaza, rampaging through southern Israeli communities and killing around 1200 people, mostly civilians.
More than 250 were captured and taken as hostages into Gaza.
Israel's retaliatory assault on Gaza has decimated the coastal territory, killing more than 53,000, according to health officials in Gaza, and displaced over two million Palestinians.
Gazan health officials have said most of those killed have been civilians but have not said how many militants have died. Israel believes it has killed tens of thousands of militants but has not provided any evidence to support those claims.
Israeli military chief Eyal Zamir on May 26 said Hamas had lost many assets, including its command and control centre.
Sinwar was elevated to the top ranks of the Palestinian militant group last year after Israel killed his brother Yahya in combat.
Yahya Sinwar masterminded the October 2023 attack on Israel that triggered the war, now in its 20th month, and was later named the overall leader of the group after Israel killed his predecessor Ismail Haniyeh in Iran.Â
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