It is unclear. Do grungs have any written language at all? And if they do, which script do they use for it?
Yes, Grung has a written form. In the PDF file One Grung Above, which details the stats for playable Grung, it lists the following:
Languages. You can speak, read, and write Grung
Since PC Grung are capable of reading and writing Grung, it must therefore have a written form. What form that written form takes, however, is never stated in any of the official material I can find - I've looked through the Grung entries in Volo's Guide to Monsters, the entry for the Grung village in Tomb of Annihilation, and One Grung Above do not state anything about what that written form looks like.
It's possible that there might be some information on this from other sources; the introduction of One Grung Above states that they were originally from the ADnD 2e books for the Greyhawk setting, and that the author of One Grung Above (Christopher Lindsay) chose to import them into the Forgotten Realms during the writing of Volo's Guide to Monsters. However, information about the Grung of the Greyhawk setting may not necessarily apply to the Grung of the Forgotten Realms.