They set off from Gainsborough, Lincolnshire, after being told the organ was available, but got stranded on the A614 in Nottinghamshire.
The county's force control room received a call from the boy's father at about 0850 GMT on Wednesday.
After being alerted to the boy's plight the helicopter was sent out to help.
Police said the helicopter unit, which is based in Ripley, Derbyshire, were routinely screening incoming calls from members of the public when they got the call from the child's father.
The team picked up the boy and a family friend while the father continued on his journey to Nottingham's Queen's Medical Centre by car.
They were flown to nearby playing fields where the boy was then taken to hospital in a waiting ambulance.
Pc Steve James said: "When we saw the call, we knew instantly that we could help in this situation and offered our assistance straight away.
"The kidney was waiting there at the hospital for him and if we did not get him there pretty quickly, the kidney would have been useless.
"The boy was fine throughout the short flight. We gave him an intercom head set so that he could hear everything that was going on - we had to make sure the little lad was happy and comfortable."
"We were very happy to help - it's one of those days where you feel really glad to have made it though the snow and into work."
the is local for me....and the goverment don't pay anything for these helicopters, all from public funding raising its shocking!!!! so glad they got to the little boy