I have played with a few different designs for dispensing tea. Some were using an auger screw to dispense tea leaves. I left the tea leaf design because I had a lot of trouble with it jamming. Tea bags are a more popular option for most people so I decided to make a design that works for that.
This module will be a standalone from the rest of the tea machine. I will probably have it controlled by its own microcontroller. I would like to have the microcontroller interface with the rest of the machine using some communication protocol (I2C, UART or other). So any other module can request for a bag to be dropped.

It consists of a continuous rotation servo that revolves a slotted cylinder. There are 10 slots which hold hold up to 9  tea bags. 

There is a gap the size of the one slot on the base cylinder of the slotted cylinder. When a tea bag in a slot aligns with that slot in the base, the tea bag will fall through the slot. Thereby dispensing a single bag.

There is a light sensor and an LED by the slot where the tea bag falls through. When a tea bag falls through it will break the beam of light and indicate that a tea bag has been dispensed.

As a module this can accept a command to DROP which starts moving the servo. Once a bag has fallen then the module will issue a SIGNAL to indicate that it has completed its operation of dispensing a tea bag. There is also a TIMEOUT command that will be issued by this module if it rotates for long enough and no tea bag has fallen. The TIMEOUT command will either indicate a mechanical fault or that the dispenser has run out of bags.