Unfortunately no. The SQL connector works because there is SQL Server transport that allows connection/transaction sharing. Because there is no equivalent MongoDB transport, in order to achieve the same thing with MongoDB, you would have to use a concept similar to the Outbox.
Currently NServiceBus built-in Outbox can be used only in the context of processing a message, not when you have a HTTP request and want to store some data and send a message. To do something like this you would have to store your outgoing message yourself and then have some sort of background worker that would push them to your transport.