Many families whose children are eligible for free school meals have not received supermarket vouchers as promised by the government because of delays in the supply system, school leaders say.

Headteachers and school business managers in England said the problem was widespread, and many took to social media to complain about difficulties accessing the scheme and delays in delivering vouchers to the families who need them.

Until now there has been support for the government’s announcement of a national voucher scheme to ensure that children from the poorest families continue to receive free meals while schools are closed during the Covid-19 crisis.

But many schools have since complained that Edenred, the provider selected by government to supply the vouchers, does not have the capacity to deal with the demand, which was exacerbated last weekend when the former education secretary, Michael Gove, unexpectedly announced that the scheme would be extended over Easter.

Although the extension was widely welcomed, there was criticism from schools, who had been given no warning and had already broken up for the Easter holidays.

“We are at risk of not being able to feed these vulnerable children,” said Stephen Morales, chief executive of the Institute of School Business Leadership (ISBL). “Emotions are high. People are tired. Families are desperate. These things need to work.”

To secure vouchers, schools have to upload data on children who are eligible for free school meals. A code is then sent or emailed to parents, who can convert it into a weekly £15 voucher for each child, to be spent in the supermarket of their choice. Many are still awaiting their codes.

Quick Guide UK lockdown: what are the coronavirus restrictions? Show What do the restrictions involve? People in the UK will only be allowed to leave their home for the following purposes: Shopping for basic necessities, as infrequently as possible

One form of exercise a day – for example a run, walk, or cycle – alone or with members of your household

Any medical need, to provide care or to help a vulnerable person

Travelling to and from work, but only where this is absolutely necessary and cannot be done from home Police will have the powers to enforce the rules, including through fines and dispersing gatherings. To ensure compliance with the instruction to stay at home, the government will: Close all shops selling non-essential goods, including clothing and electronic stores and other premises including libraries, playgrounds and outdoor gyms, and places of worship

Stop all gatherings of more than two people in public – excluding people you live with

Stop all social events, including weddings, baptisms and other ceremonies, but excluding funerals Parks will remain open for exercise, but gatherings will be dispersed.

“The problem is very widespread,” said Matthew Clements-Wheeler, director of support services at Bordesley Green girls’ school in Birmingham and chair of trustees at ISBL.

“It’s worrying our parents. They are having to ring up every day to ask, ‘Where are our vouchers?’ We are just having to say we’ve done what we can at school level. It’s just a case of waiting. We are not properly serving some of the poorest families in the country.”

Teaching unions called on the government to intervene. Paul Whiteman, general secretary of the National Association of Head Teachers, said: “Many members are having real difficulties accessing the vouchers through the national scheme. It is imperative that the DfE work closely with the supplier in question to address this matter swiftly.”

The Department for Education said it was working “tirelessly” with Edenred to resolve outstanding technical problems. “We know that the free school meal voucher system is working for thousands of schools and voucher codes are already being processed and distributed to families,” a DfE spokesperson said. “We thank schools for their patience while we expand the system to meet the high demand.”

Some schools are choosing their own alternatives to the voucher scheme, the costs of which the government has promised to reimburse. One primary school teacher in Grimsby has been walking five miles every day to deliver free school meals to disadvantaged children. Zane Powles, assistant headteacher at Western primary school, carries 78 packed lunches in a large rucksack, leaving them outside front doors before having a quick chat with each family from a distance.

After three weeks of delivering the food, he has become something of a local hero. One colourful poster put up on a fence by children read, “Well done, Mr Powles.”

For Powles, a fitness enthusiast, it’s just another part of his role. “My job is the welfare of children and educating them,” he said. “In these times I’m just doing it in a different way.”