There's no real standard way that I know of. One way could be to use spontaneous payments (e.g. keysend or AMP). All you (as the sender) need to know is the destination's public key, which I expect to be known in your context.

LND seems to explicitly support sending messages with keysend/AMP.

CoreLightning [supports](https://docs.corelightning.org/reference/lightning-keysend- keysend, but I'm not sure how easy it is attach a message.

The message is encrypted so that only the recipient can read it, so it's safe to include a way for them to respond (such as a nostr pubkey for example).

In the future Onion Messages might be used for that.

However, both OM and keysend/AMP lack a notification mechanism to alert the node operators that they've got mail. You're just sending a message to their node, with no certainty that they'll actually read it. It would be quite easy (I guess) to build a tool on top of a Lightning node that listens for keysend/AMP payments that have a message attached and forwards the message to the node operator themselves (for example via a Nostr DM). But it doesn't exist yet AFAIK.