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It seems that most languages are assumed to have a written component i.e. Common, Elvish, Dwarvish, etc. This general assumption should hold for any language inherited from the character's race. From the Player's Handbook regarding racial languages:
By virtue of your race, your character can speak, read, and write certain languages.
Certain beasts also have a language unique to their kind, such as the Giant Elk and Giant Owl. Are these languages written languages, or are they only spoken?
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 intro to the Monster Manual says this about languages in monsters' stat blocks:
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 2021: Folk of the Feywild 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.
Since Giant Elk and so on are not languages listed in the table of standard and exotic languages given in the Player's Handbook (p. 123), we have no official information to go on here.
Presumably, Giant Elk, Eagle, or Owl consist entirely of sounds that elk, eagles, or owls can make and are thus not dialects of a more well-known language. This suggests to me that they don't have a written equivalent, but your DM could certainly decide they do have some kind of representative marking system.
There are other creatures, less clearly animalistic, that have similar species-specific languages, including Otyughs, Thri-kreen, and Yetis, so there may not be a single answer to this question; it may vary on a case-by-case basis.
The lore of the Giant Elk from the monster manual is as follows:
The majestic giant elk is rare to the point that its appearance is often taken as a foreshadowing of an important event, such as the birth of a king. Legends tell of gods that take the form of giant elk when visiting the Material Plane. Many cultures therefore believe that to hunt these creatures is to invite divine wrath.
While the text of the Giant Eagle is less direct, their intelligence is clearly inspired by the Giant Eagles of Tolkien, who were servants of celestial beings. As such, it's possible that their languages might be based off of Celestial, and written using a variant of the Celestial script.
Similarly, the lore text of the Giant Owl is as follows:
Giant owls often befriend fey and other sylvan creatures and are guardians of their woodland realms.
Additionally, all three of these types of creatures are listed as potential enounters in the Feywild in Domains of Delight, and that same book notes that Awakened Beasts are common in the Feywild. This suggests that their languages may be based off of Sylvan, which uses the Elven script.
Of course, given their anatomy, it's unlikely that any of them are going to be doing much writing themselves, given the lack of hands to write with!
