Diagnosis
The latch mechanism that holds the dispenser closed is either broken or the solenoid that releases it has failed. More often than not, it's a simple mechanical jam from hardened detergent or a small piece of debris.
How to Fix It
1) Manually try to open and close the dispenser door. Feel for grit or resistance. 2) With a damp cloth, thoroughly clean the entire dispenser compartment, paying special attention to the small plastic hook that latches the door shut. 3) Run a cycle with the dispenser empty to see if the mechanism now activates (you'll hear a distinct *click* during the wash cycle). If it does, the problem was just gunk.
If That Didn't Work
The release mechanism is behind the inner door panel. Accessing it requires removing the door's inner liner, which is a more complex disassembly. The plastic latch parts are known to wear out or break.
Heads Up
Don't use liquid detergent in the main dispenser if you can avoid it. It leaks out and gums up this mechanism faster than pods or powder. If you must, put it in the pre-wash cup only.