How do I add a commercial coffee maker to my fridge's water line?
There's typically a solenoid valve right down at the bottom where the supply line from your house connects to the fridge, and this valve is activated by the ice maker when it needs more water.  

Unfortunately, you've tapped in half-way up the back of the fridge - after the solenoid valve - so as a result, your coffee machine is only going to receive water for the brief period that the ice-maker in your fridge refills in between ice-making cycles.  

You need to put that coupler back where you found it and instead find a way to tap into the water line which supplies your fridge from the wall.