The Trussell Trust said it was on course to deliver a record number of food parcels from its 428 food banks across the UK during the current financial year.
Food banks in areas in which UC has been introduced for at least six months have seen a 30% increase in demand compared to the year before, according to the charity.
The figure was branded "shocking" by shadow work and pensions secretary Debbie Abrahams, while Oxfam said it was "increasingly concerned" by the effect of the UC roll out.
A spokesman for the Department for Work and Pensions (DWP) said it would be "misleading" to link food bank use to any one issue.
The Trussell Trust said the six-week wait for UC should be cut as a matter of urgency, and called for administration of the benefit to be improved and reassessment of the freeze on benefit levels.
Mark Ward, interim chief executive, said: "We're seeing soaring demand at food banks across the UK.
"Our network is working hard to stop people going hungry but the simple truth is that even with the enormous generosity of our donors and volunteers we're concerned that food banks could struggle to meet demand this winter if critical changes to benefit delivery are not made now."
Ms Abrahams, Labour MP for Oldham East and Saddleworth, said: "The shocking 30% increase in food bank use in areas where Universal Credit has been fully rolled out is further evidence of the desperate need for a pause to the programme while it is fixed.
"As well as foodbank use, the six-week wait for support and cuts to the programme are also driving debt, arrears and even evictions.
"The social security system is supposed to prevent people from going hungry and getting into debt, not make these problems worse."
The Trussell Trust revealed it had distributed 586,907 emergency supplies in the six months to September, over 67,000 more than during the same period last year.
More than 200,000 supplies went to children, while on average, during the past year, people needed around two parcels each.
Oxfam's head of UK poverty, Rachael Orr, said the figures were "probably just the tip of the iceberg".
She added: "Oxfam works with many more independent food banks who are seeing a similar rise.
"We are increasingly concerned that the way Universal Credit is being rolled out is causing unnecessary hardship.
"The Government needs to urgently rethink and cut the six-week wait."
The Archbishop of Canterbury, Justin Welby, said efforts must be made to ensure foodbanks do not become "a staple necessity".
He said: "Figures on food bank use in Britain are utterly shocking.
"Many households in the UK are just one crisis away from needing assistance of this kind. It is a scandal that in 21st century Britain we have need of them at all."
A DWP spokesman said: "The reasons for food bank use are wide and complex, and for this report to link it to any one issue would be misleading.
"We're clear that advance payments are widely available from the start of anyone's UC claim, and urgent cases are fast-tracked so no-one should be without funds.
"We know the majority of UC claimants are confident in managing their money. Budgeting support and direct rent payments to landlords are also available to those who need them."