I play D&D since 3.0 came out and never encountered this decided in rules. D&D Beyond search is a closest thing to complete and objective query I can think of to try to prove a negative.
A search on D&D Beyond reveals no entries for “giant owl” or “giant elk” that would say anything about written form, or lack of it, of their languages. Assuming D&D Beyond is reasonably complete, and its search is accurate, the only thing we can say is that at the moment rules doesn’t say one way or another.
It’s worth noting that languages meant for player races have written forms (per the quote in the question), but the Monster Manual says:
The languages that a monster can speak are listed in alphabetical order. Sometimes a monster can understand a language but can’t speak it, and this is noted in its entry.
Nothing about writing there.
Owlfolk, a player race from Unearthed Arcana which are described as cousins of Giant Owls, do not have Giant Owl as racial language, but that proves nothing. If they did, then the language would have to have a written form, but it seems the author was careful not to make this decision.
TL:DR For now, DM decides.