Mr Williamson said that while some schools were making their own plans to ensure pupils continued to be fed, he hoped most would join the scheme.
He said that no child should have to go hungry as a result of measures brought in to control the spread of the virus.
“I really hope that this is something taken up by most schools. I think this will be a real big help to so many families,” he told the PA news agency.
“We are living in extraordinary times at the moment. It’s really important that we do everything we can do for families, make sure those children we really want to be able to support and help are getting that support.
“What we are wanting to do is just give schools that extra bit of support, give families that extra bit of support.”
The Government originally announced the voucher plan following the decision earlier this month to order the closure of all schools.
The vouchers can be spent on food at a range of shops including Sainsbury’s, Tesco, Asda, Morrisons, Waitrose and Marks and Spencer, although Mr Williamson said he hoped to get all the supermarket chains signed up.
Once a school has joined the scheme the vouchers will be sent out by email directly to the eligible families by the Department for Education’s supplier, Edenred.
Alternatively, for those families which do not have internet access, the school can claim them on their behalf and send out paper copies in the post.