Have you tried with that tutorial?
If you want to use Yocto, you will need to use V4L or ffmpeg for this, do you already have a webcam?
You should be able to run Arduino sketches in your platform, I’m assuming the image you want to use is Ubilinux, if this is the case I suggest you to contact EmutexLabs for more information about this image and it features.
You can use the UART of the board to send information to another device. You can use the pins 0 and 1 in both boards in order to use the Rx and Tx.
I have ordered a USB UVC webcam but have not received it yet, it should be here this week.
I wasn't sure what image to use but Ubilinux looks good so I'll try that one.
What could I use for the web server that will work with Ubilinux and be able to use processing.js on the page?
The supported image is Yocto, there are two versions: the uClibc and EGLIBC, you can download both from: IoT - Intel® Galileo Board Download | Intel® Developer Zone
If you want to change to the Ubilinux image you can, I suggest you to contact EmutexLabs for questions while using that OS.
If you want to use Yocto, as I said you can create a web server using V4L, FFMPEG or OpenCV [Python, Node..] there are some threads related to this in the community that may help you on this.