Can an encrypted message be derived from only keys? I recently started teaching myself cryptography, and had an idea while reading up about basic ciphers. Instead of passing ciphertext between Person A and Person B, which can be broken if intercepted by Person C, what if they passed keys between them for a ciphertext they both possessed with multiple messages encrypted in it? If Person C intercepted multiple keys, could they derive the ciphertext from them?
No, you cannot derive the plaintext from keys if the plaintext nor the ciphertext is ever transmitted. However, to break a cipher you don't need to do that. You only have to show that you can get any kind of information about the plaintext (other than information about the message size, which is hard to avoid if any size message can be present).

Now if you see repeated "keys" then you know that the same message is being targeted, and the scheme would be considered broken. This is why ECB mode is considered broken: you cannot decrypt any message, but you can see repeated blocks.

Now in your case you don't seem to have any cryptographic keys, as keys cannot be shared with everyone - what you seem to be doing by sending them over a unsecured channel. Your lookup scheme doesn't really use cryptography in that sense. They may be database keys, but database keys are a different concept.