Not with current implementation, unfortunately.
You can lift the mirror and keep it lifted with a piece of tape; however, the mechanical shutter is actuated. This part happens on a different CPU that is not very well understood (
the MPU). You can take pictures with shutter open (see
FRSP), but has a
gradient at
short exposures. Additionally, file saving and image capturing must be pipelined to achieve 4 fps.
If resolution and rolling shutter are not an issue, you could look into
half-shutter triggers for raw recording. However, its timing is given by the LiveView clock, not by the external trigger.
Delaying pictures in regular burst mode using an external signal might be easy to implement (didn't try, but I don't see why it wouldn't work). Not sure if half-shutter can be used as a sync signal though, as it must be pressed to keep the burst running.