Sullivan's personal donation will go towards Rashford's campaign through charity Fareshare UK and the Child Food Poverty Taskforce, which was founded by the Manchester United striker.
The Hammers joint-chairman, who was worth £1.15billion in 2019 according to the Sunday Times rich list, told the club's official website: "Marcus Rashford has done such a fantastic job of raising awareness and leading the way in tackling this very important issue, which impacts so many youngsters in our local communities up and down the country. It is a privilege to be able to help.
"On a national level, the impact of the Child Food Poverty Taskforce will be huge. While closer to home, I’m really proud that the club has been supporting free school meals during school holidays for years now and we have seen what a continued difference it is making in our local community."
West Ham's neighbours Millwall have also joined the campaign, with the club's Community Trust and the Millwall Supporters' Club uniting to provide 100 hot meals per day for children during the October half-term.
The project builds on the Millwall's summer programme, in which The Lions and its Trust provided 5,000 meals over the school holidays for children and their families.
"The work of the Community Trust, which we have witnessed first-hand, has been truly humbling," said Tom Gale from the MSC.
"When an opportunity arose to unite the fans and build on the already successful summer programme, we were more than happy to help. With national attention on this important matter, Millwall's fans have once again stepped up to the plate and shown how much they are willing to help the most vulnerable in society.
"As the official supporters’ club, we are proud to represent them."
Elsewhere in football, Rashford's Manchester United have announced they will provide 5,000 meals for vulnerable children over the October half-term holidays and Leeds United has said it will match the £25,000 donation pledged by its squad to the campaign.
Councils, restaurants and cafes have been among the thousands of organisations to support the scheme by offering free school meals to vulnerable children after a Labour bid to extend the vouchers into the holidays was rejected by MPs in the House of Commons last week.