Church Action on Poverty and Oxfam said the Government was failing to properly monitor the numbers now resorting to emergency help.
The Commons work and pensions select committee should launch an urgent investigation into "the relationship between benefit delay, error or sanctions, welfare reform changes and the growth of food poverty", they suggested.
Up to half of those seeking help were doing so as a direct result of having benefit payments delayed, reduced or withdrawn, their report said.
Energy Secretary Ed Davey told MPs this month that it was "completely wrong to suggest that there is some sort of statistical link between the benefit reforms we're making and the provision of food banks". However, the charities drew a direct link with the Government's two-year below-inflation cap on working-age benefit increases.
The report, Walking The Breadline, concluded: "There is clear evidence that the benefit sanctions regime has gone too far and is leading to destitution, hardship and hunger on a large scale. There is a real risk that the benefit cuts and the introduction of Universal Credit ... will lead to even larger numbers being forced to turn to food banks. Food banks may not have the capacity to cope with the increased level of demand."
The UK's biggest provider of food banks has reported a trebling of the numbers using them to more than 350,000. But the charities said at least half as many again were receiving help from banks run by organisations other than the Trussell Trust.
Oxfam CEO Mark Goldring said: "Cuts to social safety-nets have gone too far, leading to destitution, hardship and hunger on a large scale. It is unacceptable that this is happening in the seventh wealthiest nation on the planet."
Niall Cooper, Church Action on Poverty CEO and the report's lead author, said: "Food banks are not designed to, and should not, replace the 'normal' safety net provided by the state in the form of welfare support."