With zaps there are two nostr notes published, one by the payer (containing the recipient user's pubkey & amount), and one by the recipient lightning node (containing the invoice and preimage). NIP 57 notes that the latter is not really proof for 3rd parties, but the former would be sufficient for the recipient to prove who requested the invoice.
Note that of course these are nostr protocols, not LN, but it should give you an idea of what you can do with LN.
https://github.com/nostr-protocol/nips/blob/master/57.md