In computer programming there's something known as the metadata, which is commonly defined as "data about data".
Thence I'd logically conclude that a transporter can beam itself, since this is explained by the metadata handling lesser units of data.
Recursion is more aligned with this thread.
How the hell can a dissociated cloud of atoms function as a transporter?
Becuase the atoms are in a superposition of many possible states and entangled with each other therefore no reason to assume physical laws can't continue in the matter stream in a sense.