When all other meesages sent by TB end up in the Sent folder than those that are generated by a mailto: link should get there too.
In case you've got multiple accounts, it's generally the same account that's used in this case. One account should have the setting 'This account is the default for "mailto:" URLs' enabled and that's the account where you need to look for the sent messages.

Of course you won't have a local copy when you filled a webform, as those won't be sent by TB and when doesn't know about the message, it won't have a local copy either.