This is a common issue for Canon users trying to use an external flash. This is actually a limitation of the camera, described in the camera manual (most models):

Cautions for Live View shooting
A non-Canon flash will not fire during Live View shooting.

Dragonframe provides a workaround. In the Cinematography workspace set External flash to On. Dragonframe will disengage the live view prior to capturing the high-resolution image, allowing the external flash to fire.