A cellphone with camera is quite serviceable as an instrument for recording video or photos of accidents. It's small enough to carry in a pocket.

Ordinary point-and-shoot compact cameras typically record video, as well as photos. You can find handlebar mounts for cameras on eBay. They attach to the tripod mount socket on a camera and clamp onto the handlebars.

The cheapest "action" camera I've seen is the Muvi VCC-003 aka AEE MD-80. It's often discounted to about $65. It's strictly a video cam, no still photos.