I know for a fact it will work and barring a sudden change in the rules this year it should be legal.
We put a dual core Gateway netbook on the robot packed in foam and drove around several competitions with it powered on connected to USB cameras.
It had an SSD in it and that year was Ramp Riot. So we drove over that bump in the field and took all the expected hits and it still worked. No we didn’t put it on a robot and fall off last year’s pyramid but I doubt it would be all that much more risky.
There are plenty of adequately powerful systems that could be put on the robot within the $400 cap. The Classmate as others have said might be by comparison a bit slow but strip down everything and run Linux and it’ll work. My older Dell Mini 9 even worked with a bone stock install of Ubuntu 9 and was the first netbook we considered for this purpose. Then we realized we could do better with a little money used.
I would advise picking a USB camera and not the integrated camera. Depending on how you go about this you’ll discover the integrated white balance and automatic whiteout protection can be an issue with these CMOS cameras. You really want to be able to develop around a camera you can rely on and then you can build on that over time. I’ve got a small plastic crate full of USB cameras from different experiments. Cost is NOT a good way to know you’ll get a good camera. There are some $100+ USB cameras in that crate that are very bad for this purpose. Oddly enough there is $2 USB camera in that crate from Amazon (they don’t sell it anymore or I’d link it) that works okay.
I suggest you start off by looking at game system cameras for the XBox and PS3. After all big companies know they will be using these for recognizing images, colors and movement. Plus with the XBox One and PS4 these are surplus.