Home for Good, which works around adoption and fostering, said it wanted "concrete plans" from the prime minister after he told MPs the UK would re-settle a "specified" number of children.
Mr Cameron said he would no longer seek to overturn a Lords amendment to the immigration bill which requires the UK to take unaccompanied children who have made it to Europe from Syria.
Speaking at Prime Minister's Questions, he gave no indication as to how many children who had succeeded in reaching Europe could now be admitted, although officials indicated the numbers would run to "more than tens".
While Mr Cameron said ministers were holding talks with local authorities and the charity Save the Children about what assistance they could offer, he stressed he would not take any steps which would encourage more people to make the dangerous journey across the Mediterranean.
Home for Good said it had more than 10,000 people respond to its call for families willing to foster a child refugee.
A spokesperson said: "Home for Good regards David Cameron's statement today about welcoming unaccompanied children in Europe as a positive development in response to on-going pressure and demand from the public and charities.
"Over 10,000 people responded in September to Home for Good's call for families willing to foster, clearly demonstrating a desire to help and support some of the world's most vulnerable children.
"Home for Good is keen to see concrete plans and timescales as soon as possible, as the need to protect these children is urgent.
"The charity will be encouraging existing foster carers to contact their Local Authorities to express their support for the Government's statement."
Labour peer Lord Dubs - a former child refugee who fled from the Nazis and who tabled the Lords amendment - said it would ease the plight of some children.
"I trust the Prime Minister will be true to his word and move swiftly to ensure the Home Office works closely with local authorities to find foster families to give these young people a stable and secure home," he said.
David Cameron said that the scheme was aimed at re-settling lone children who had reached Europe before the EU's deal with Turkey in March which effectively closed the migrant trail through the Balkans.
He strongly defended the Government's approach in concentrating its main effort on supporting refugees still in the region in countries such as Turkey, Jordan and Lebanon.
"I do want us to proceed with as much support across the House as we can," he told MPs.
"What I don't want us to do is to take steps that will encourage people to make this dangerous journey because otherwise our actions, however well-meaning they will be, could result in more people dying than more people getting a good life."