North Yorkshire Fire and Rescue helped 26 Easingwold School pupils off the vehicle when it became stranded near Newton-on-Ouse just after 08:00 GMT.
None of the children was injured in the incident.
North Yorkshire County Council said the bus driver went through a road closure sign and it would be investigating.
In a statement it said the route taken by the bus, operated by Stephenson's of Easingwold, had been closed due to flooding.
The bus operator apologised and said it was carrying out a detailed investigation.
"We put road closure signs out during flooding for a very good reason - to keep people safe," said Councillor Don Mackenzie.
"It is not possible to tell how deep the water is on flooded roads and drivers can quickly find themselves in trouble, along with their passengers."
Dave Stephenson, from the bus operator, said: "We have yet to interview the driver but obviously as soon as we speak to him we will know more about what's happened.
"We express our sincere apologies to the children who were on the service this morning as well as to their families for this unfortunate incident."
North Yorkshire Police said they would be investigating how the bus became stranded on the eight-mile (12km) journey to the school.
Head teacher Phil Benaiges said: "A number of students were brought into school but most went home with their parents to get warm and dry.
"Some were upset and all were cold and wet."
River levels remain high following flooding over the Christmas and New Year period and around 15 roads across North Yorkshire remain closed.
The fire service and police said it was essential motorists did not attempt to drive through flood water.
Further heavy rainfall on Monday virtually cut off the seaside town of Whitby where the fire service had to rescue two people from a car which became trapped by flood water.
A bus carrying almost 30 school children became stuck in flood water near York after the driver allegedly ignored a road closure sign.