The lightning struck in several areas of the Vieux Canal festival in the town of Azerailles, the regional council said in a statement.
Those injured include children who were in a tent during the storm.
The victims were "directly hit by the lightning and suffered burns", the regional council said.
A woman in her sixties and a 44-year-old man are reported to be in a serious condition as a result of the strikes.
Parts of France around this time of the year are often put on alert for violent storms.