I assume that this is possible based on the fact that both utilize RSA for encryption. I should be able to read X.509 and store it as a new OpenPGP key.
The collaborator of my software needs OpenPGP. Another collaborator provides X.509.
I am looking for a way to convert the keys. Is that possible, how would one do that?
The short version: you can somewhat, but there is rarely good use in doing so.
You can extract the numbers forming the key and theoretically put together a new X.509 and/or OpenPGP key from them, but those would still remain incompatible, different keys in the respective system. Actually, the monkeysphere project brings tools for both directions (openpgp2pem and pem2openpgp, but make sure to read the rest of the post before heading out and converting keys).
Both X.509 and OpenPGP are more than a file format for keys: they add (incompatible) options for key management and certification, metadata, identifiers, ... Also, both systems use slightly different cryptographic modes of operation, and have very different (and thus incompatible) formats for encrypted and signed messages. They even have enormous differences in how certifications are handled (hierarchical structure in case of X.509 vs. an arbitrary graph in case of OpenPGP).
With other words: anything you do with the X.509 "representation" of an OpenPGP key sharing the same RSA primes cannot be used with the OpenPGP variant, and the other way round. Certificates issued in one system don't work in the other (and cannot be converted!).
As both keys "representations" are incompatible anyway and have to be managed separately, I would strongly recommend to create different sets of keys from beginning. After all, this adds another layer of security in case one of the keys is breached, as the other key stayed undamaged. Apart from performing unusual operations is always error-prone and suspicious to follow-up issues.
There might be good use cases, for example the monkeysphere project requires those conversions for authenticating SSH connection through OpenPGP keys. But I would not consider general usage for signing and encrypting messages and files a good use case for the reasons given above.