I would now like to add a function with a loop that runs until another string comes in from the Ethernet controller, is this possible?
You have that already. It's called loop().
If are planning to wait for another string from the same client, you are going to be disappointed. The client has made a request and gone away. If another client request appears, you have no way of knowing that it is the same client you hear from last time, or another client on the other side of the world. Nor should you try to know that.
I want to create a animation that runs without running through the rest of the code
Have you looked at what the client is actually sending? Each GET request is standalone. The same client might make a series of requests, but the server has no way of knowing that it is the same client.