It will need internet connectivity briefly, then it will encrypt everything even if you disconnect.
As soon as CryptoWall runs, it will generate a random RSA public and private keys, connect to one or more control servers, and upload the private key along with some information on the system it is, like OS version, public IP and location. Then it can run disconnected.
After generating the keys, the malware will proceed to encrypt every single file it supports. Some variants will keep the original intact during the encryption phase, and delete all original files after encrypting all of them. Some variants will encrypt and delete the original in one pass.
The server will keep the private key for some time, and you have to upload a sample of an encrypted file to them. They will send back the private key you will need to decrypt the files.
You probably have no chance of stopping the encryption process once it started. CryptoWall employs a number of defenses, and if you get caught by it, it's over. The encryption is well made too, so the chances of bruteforcing it is almost zero. Backups will not save you if your backup drive is connected to the PC all the time, as CryptoWall will encrypt your backups too.