Where is the blend door actuator located?

Also, I cannot hear any clicking noises or anything else that would indicate a problem.

Hello,

It does sound like a faulty blend door actuator. It is probably just stuck in one position and not turning at all that's why you do not hear the typical clicking sound.

To access the blend door actuator you will have to remove the radio and all the trims below the radio.
Your vehicle has 2 actuators, one for driver and other for passenger side.
See diagrams for the location reference.

https://www.2carpros.com/articles/replace-blend-door-motor

Let me know of any questions.
Thank you.

Does the radio need to be removed?

Yes, you will have to remove all of the center console/dash area. Look at the picture below.