X.509 is a format for certificates: a certificate is a sequence of bytes which contains, in a specific format, a name and a public key, over which a digital signature is computed and embedded in the certificate. The signer is a Certification Authority which asserts that the public key is indeed owned by the entity known under that name. By verifying the signature, you can make sure that the certificate is genuine, i.e. is really what the CA issued; and, that way, you gain trust in the binding of name with public key (insofar as you trust the CA for being honest and not too gullible, and as you know the CA key for signature verification, which may entail obtaining a certificate for the CA, and verifying that certificate, and so on, up to a trust anchor aka root certificate).
Thus, X.509 is a way to distribute public keys, by which I mean: a method which allows various actors (e.g. you) to know, with some guarantee of non alteration by malicious third parties (i.e. "attackers") the public keys of other actors.
OpenPGP is a standard format for a lot of things. One of the things which OpenPGP defines is a way to encode a public key along with a "name" (an email address), and a signature over these two. That's, really, a certificate in its own right (although with a format which is not compatible with X.509). But OpenPGP also defines how to use the public key of a given individual (let's call him Butch) in order to encrypt a bunch of bytes that only Butch, using his private key, will be able to decrypt. Technically, this uses a randomly generated session key, used with AES (or similar) to encrypt the raw data, and that session key is what is encrypted with the recipient's public key (generally of type RSA or ElGamal).
Therefore, for your problem, you do not want to "encrypt with X.509". X.509 defines nothing about encryption. What you want is to use a standard format which describes encryption with the recipient's public key and builds on X.509 certificates for the public key distribution. This standard format, coupled with X.509, would be an analogous to OpenPGP. This standard format exists and is called CMS (formerly known as "PKCS#7"). When CMS objects are sent by email, this becomes another layer of standard, which is called S/MIME.
CMS (or S/MIME) is what you need for asynchronous communication: you prepare a blob, to be sent later on to the recipient, like what you do with OpenPGP. If you can make a synchronous communication (sender and recipient are "online" simultaneously), then you can use SSL/TLS (or its Web counterpart HTTPS). In SSL, the server has a public key, which it sends to the client as an X.509 certificate. The client validates the certificate, then uses the public key contained therein in order to establish a session key with the server, and encrypt the data with that session key.
In any case, assembly of cryptographic algorithms into protocols is known to be hard to do correctly, nigh impossible to test for security, and fraught with perils. So don't imagine that you may build your own mix; you really should rely on an existing standard, like CMS or SSL.
