It'd need a fair bit of work on both BitCoin and the network. Chrome[OS] only allows WebSockets, not raw sockets. If port 8333 in the standard software accepted WebSocket connections as well then theoretically you could run a version of the regular software that had been NativeClient-ified.
Hosting your wallet on some remote server like MyBitcoin or equivalent service is probably the way to go for ChromeOS. It's an OS designed for the cloud so trying to run something very complex and resource intensive on it won't work well, the hardware just isn't high powered enough. Using a cloud wallet is much more in the spirit of the device.