Should I use independent symmetric keys for each client should I use keys derived from a master key?
Yeah, sure, you can derive keys from a master key. This is often performed for schemes using symmetric encryption primitives. And yes, including the ID of the client is generally the way to do this.

SHA-256 is not a brilliantly secure KDF - but in general it will do. You'd better use a real Key Based Key Derivation Function though. Currently HKDF is all the rage, which uses HMAC underneath. HMAC in turn can be configured using a secure hash such as SHA-256. 

When starting from a secure master key you can directly use HKDF-expand (skipping HKDF-extract) where available.



Note that transport security requires extra care. In general you may want to bold on perfect forward secrecy (ephemeral Diffie-Hellman) or other key derivation techniques that create session keys from your client keys. You may also need a form of authenticated encryption using HMAC or an authenticated cipher such as AES-GCM.