Yes you can record video using EDSDK, refer to Section 6.4 in the EDSDK Manual (the one that comes with the latest v 2.13.20)
e.g. C++ example from that page for starting/stopping video:
EdsUInt32 record_start = 4; // Begin movie shooting
err = EdsSetPropertyData(cameraRef, kEdsPropID_Record, 0, sizeof(record_start), &record_start);
EdsUInt32 record_stop = 0; // End movie shooting
err = EdsSetPropertyData(cameraRef, kEdsPropID_Record, 0, sizeof(record_stop), &record_stop);
If you want to use Java, please look into some of the Java wrappers flying around, e.g. https://github.com/kritzikratzi/edsdk4j . I am not sure if these are up to date and reflect all the necessary commands.