Can I store my wallet on one machine while running the client on another? I'd like to help the p2p aspect of the network by leaving a dedicated machine connected...
Here's how I understand your question: You have a spare computer that you'd like to run bitcoin on to help the network. You have a second computer that is your main personal computer; it's also where you'd like to keep your bitcoins. Two copies of the blockchain would waste bandwidth and disk space.
There is a way to keep the blockchain on one computer but to keep your coins on another - you can run the electrum client, and the normal bitcoin client on the other computer. Optionally, you can run the electrum server on your always-up computer, although this requires some command-line knowledge.
The wallet functionality and peer-to-peer node are not very tightly dependent, a node does not require a wallet to store the chain, validate and propagate transactions.
A wallet is basically just a keystore (and a list of available unspent outputs).
Your question is not very clear though, do you want to leave your wallet on another machine for security purposes? In this case you might want to switch to a wallet like blockchain.info which might be more practical.