Shadow education secretary Kate Green said now was “the time to act".
More than one million children could go hungry over school breaks because of the coronavirus crisis, she warned.
Mr Johnson bowed to public pressure, including from some Tory MPs, over a previous campaign by Mr Rashford which secured free school meals for thousands this summer.
But No 10 is resisting pressure to change its policy this time around.
A petition launched by the Manchester United striker secured 100,000 signatures in just 10 hours.
The Welsh Government has pledged to provide free school meals during the holidays until Easter.
But a Number 10 spokesman said: "It's not for schools to regularly provide food to pupils during the school holidays."
Ms Green said: "Millions of families face the prospect of losing their livelihoods because the Government has lost control of the virus.
"Now is the time to act. Labour will not stand by and let families be the victims of the Government's incompetence.
"If Boris Johnson doesn't change course, we will force a vote … and give his backbenchers the chance to do the right thing."
Rashford was recently made an MBE for his services to vulnerable children.
His petition calls for free school meals to be expanded to any household on Universal Credit, as well as provided during holidays.
Geoff Barton, general secretary of the Association of School and College Leaders, said: "Schools are working incredibly hard to help children catch-up with lost learning amidst ongoing disruption caused by rising Covid infection rates, and the pupils who need the greatest degree of support are often those from disadvantaged backgrounds.
"To then have a situation where they are potentially going hungry through holiday periods is very obviously detrimental to both their welfare and educational progress."