Depends on the drawer. First check to see how much clearance you have from the closed door to the front of your drawer.  Then pull out your drawer and measure how much clearance the drawer has to the open small door.  Some drawers open very close to the small door and those won't have enough clearance, others have several inches of clearance to the small door and those can work.
They have 3 options for the small door (not including Nadir's custom larger pocket):
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If you have very little clearance to your drawer, perhaps the MOLLE version would work, then you could choose thinner MOLLE pouches or other gear to hang on it.
The Overland Outfitters Grenadier products are here FYI: 
https://www.overland-outfitters.com/ineos-grenadier.html#/.   You can also message them directly from the web site with any questions.   And they ship worldwide.  And BTW they also have a version for the wide door, and this photo shows the narrow door with the MOLLE version and a MOLLE pouch attached:
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