Lloyd Mason-Edwards from Peel Park Primary School in Bradford, said only about half the families eligible at his school had received vouchers.
He said the system was a "shambles" and was failing those in need.
Edenred, which runs the voucher scheme, said it was aware of problems and that many had successfully claimed vouchers.
The government has said it will provide extra funding to schools to cover such "unavoidable costs".
Staff at Peel Park school are making up parcels of food and toiletries and Mr Mason-Edwards delivers them to 10 of its most vulnerable families each week.
He said of the 650 pupils at his school, 150 qualified for free school meals and about 10 of those were really struggling.
"They [the government] are not providing for the most vulnerable members of our community," he said.
"The ones that don't speak English, or the ones that have been furloughed and are yet to receive money.
"We have some parents that started up a business five months ago and are not receiving any payments either."
The government scheme to give poor pupils in England vouchers worth £15 a week until schools reopen has experienced numerous problems.
Mr Mason-Edwards said the vouchers were provided by email so those without internet access were "slipping through the net".
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