Is there a cryptographic hash function that can be performed with pencil and paper?
I don't think doing this by hand is the right way to do this. Pencil and paper systems are unreliable because humans just aren't as good at being robots as, well, robots! Most of the work in computing a secure MAC is hugely monotonous and a single mistake, anywhere during the process, could completely destroy any security you thought you had.

If your library computer has web access, then it executes Javascript with very high probability. Javascript is enough to conquer this problem.

Just create a static HTML with some Javascript in it. You have two HTML input fields with a single button. When you click, it computes the $HMAC(site,password)$ using SHA-256 or something, then encode the result as printable characters.

There is the problem of key loggers etc, grabbing your master password but they could just as easily covertly film you as you carry out the computation by hand!

There's a limit to the countermeasures that be deployed when the machine is fundamentally untrusted. However, you can mitigate against key loggers by simply making the master password be entered, character by character, via a dropdown box.