The good: I think I have a good enough grasp on mjpeg-streamer and the Bridge library to build some controls. My first test page showed the analog readings off of A0-A5 and a stream output of the microscope.
The bad: While testing this, I had hot-wired the uArm-shield to back power the Yun by jumpering the 5V line to the Vin line. This worked fine until I started streaming. When streaming from the camera, the light on the microscope was flickering, and I soon found out why: The regulator on the uArm shield, an NCP4625, is only rated to 300mA. I was asking way more of it than that while running the streaming output. I feel sorry for my Yun, for starving it for power for so long.
The fix: The large ethernet port and usb port on the Yun causes me, and as far as I can tell, many other people minor issues. I decided the best way to fix all these problems was to build a regulator mezzanine specifically for the Yun that will let you power it from a 5V wall-wart through a barrel jack. My results can be found at https://oshpark.com/shared_projects/IkCdJgfk.