(sorry for the delay as we are just back from vacation)
I am not sure how this would work with the Arduino WIFI shield. Video compression is no problem using OpenCV or MJPEG, for example see how JeVois can stream compressed MJPEG video out of its USB port, or how we have used the OpenCV VideoWriter to save compressed video to microSD in the SaveVideo module. The part that may be tricky is how to transfer the (still large) amounts of streaming video data to the wifi shield. From my understanding, the Arduino communicates with the shield over SPI, but you would need a way to communicate the video data over USB between JeVois and the shield.
You can do this with OpenWRT boxes since they have a host-USB port to which you can connect JeVois as a webcam. Indeed those are quite slow though, we could barely achieve 15fps at 320x240 in our tests.
Have you looked into the Raspberry Pi Zero W? This might be a good option, faster than typical OpenWRT boxes. You would still use mjpg-streamer or similar to grab video from JeVois connected to the ZeroW as a USB webcam, and to transmit that video over the Wifi link.