Is a ham radio license required to fly a balloon?

No, there are no licenses required in order to fly a balloon, however a license may be needed depending on how you are tracking your balloon.

One of the more popular ways to track a balloon is by using a ham radio protocol called APRS (Automatic Packet/Position Reporting System).  This protocol allows for a tracker placed on the balloon to transmit its position and telemetry data in real time to the ground.  A vast network of ground stations receive this data and will "gate" it onto the Internet for both you and others to see.

In order to use this APRS network, someone on the team will need to be licensed with at least a Technician Class Amateur Radio Operator License in order to use the APRS network.  Only one license is required though, as anyone with the right receiving equipment, or even an Internet connection can track the position and progress of the balloon.