Need to check the actuator motor. Drivers side under the dash, towards the center console. Do you hear any movement in that area when you select a different vent position?

 

I do not hear any movement in that area when I select different vent positions. How do I go about fixing this?

 

If your sure the actuator in not moving.

 

I just had this same problem. Will the air come out anywhere else? If not it probably is the mode motor(87106-06060) as someone else said.

 

Does your Solara have automatic climate control? (SLE models have the auto air) The servo that controls the 2 dampers that would relate to your issue is the AIR MODE servo which is above the climate control servo, both servo's are drivers side. If you remove the plastic panel below the steering wheel (panel with the hood release) Stick your head up there to the right of the steering wheel you'll see both servo's.
Run the car and adjust the controller for different settings and you should see the top servo moving when you switch modes, the bottom climate control servo will move when you move the temperature setting. In my experience these servo's are very quite, when you hear them move is when they're are about to crap out.

 

I have a Solara SE (1999). I'm not positive that the actuator isn't moving. I can't hear it move when I rotate through the different air options. When I switch the different air options now it only comes out through the front. The air doesn't come out anywhere else.

 

I just checked mine there should be about 180 degrees of travel when switching modes. (full air at bottom vents versus defrost) With a flashlight you can easily see the servo without dismantling anything. Run the car stick your head under the steering wheel and operate the mode control with your right hand.
It looks just like the one on the video that patcal096 posted. You should verify it's moving correctly, it may not be moving fully. It's operating 2 dampers at the same time as you switch modes.