Well, the less painful way (And MAYBE cleaner) would be to have your pickup script for the disk set a global variable to 1 after the player picks it up. (Make sure this variable isn't being used by any other scripts except your trigger zone and the disk script)
Now, you'll need to make a new story zone script... just copy and paste the one that the storyzone already uses and edit the copy. In the line that checks if the player is in the zone, have it also check the global variable for a value of 1. If the variable is set to 1, then you can be pretty sure that the player has picked up the disk and can therefore play the video.
The NON-variable way would be to make a NEW story zone marker and edit it's FPE file to have a "UseKey" property set to the name of your disk. Then make a new story zone script and have it also check for a "HaveKey=1".
There you are... in a nutshell that's how to do it.
If you are unfamiliar with scripting with variables or creating new marker entities or ANYthing above... I suggest that you read my tutorial on using the V109 variables (unless you're using Ply's Mod, in which case there is no tutorial for those variables and I suggest you learn scripting front and back for using the Mod anyways) and look around the boards a little bit for any other info you may be confused about.
The one and only,
Those who live in the past, are destined to insanity. Those who live only for the future, will be slaves to their ambitions.
Those who live in the moment... only they, are truly happy.