I have been unable to find this anywhere and was curious about it. I've found the relationship between all other languages and which script, if any, they have. The Aarakocra language seems to have a script, since it is listed as you can write in it, yet I can't find the script itself, whether it be it's own or a shared script like Dwarvish.

I've checked the Elemental Evil where they were introduced, but it doesn't mention their script. And I heard there was something about Aarakocra in Volo's Guide to Monsters, but I can't find any mention of them, let alone their language.

Maybe Dwarvish or Draconic

I had a search through resources for 5e and previous editions, but couldn't find mention of the script used for the Aarakocran language. In fact, Aarakocra doesn't seem to have existed as a language in its own right before the publishing of the EEPC; according to their original entry in the 5e Monster Manual, they only speak Auran, with no mention of a native language of their own. In 3.5e they are noted as speaking Common and Auran.

Aarakocra seem to be generally described as being dwellers of the Elemental Plane of Air (where you'd expect Auran to be the primary language) and are all fluent in Auran already. It seems plausible that the Aarakocra language is derived from Auran - and Auran (in 5e, at least) is itself a dialect of Primordial, which the PHB states uses the Dwarven script, so maybe Aarakocra does too.

Alternatively, in prior editions the different elemental languages were not so related to each other, and they used different scripts. 3.5e describes Auran as using the Draconic alphabet, so that could also make sense (maybe the draconic script is easiest to write using talons).

In 5e, the rules don't give a straight answer to your question, but Aarakocra most likely uses the Dwarvish script.

Aarakocra are described as originating from the Elemental Plane of Air:

They hail from a world beyond—from the boundless vistas of the Elemental Plane of Air.

_{Elemental Evil Player's Companion, p. 3}

Furthermore, they "speak, read, and write Common, Aarakocra, and Auran".

Auran is described as a dialect of Primordial:

Some of these languages are actually families of languages with many dialects. For example, the Primordial language includes the Auran, Aquan, Ignan, and Terran dialects, one for each of the four elemental planes.

_{PHB, p. 123; emphasis mine}

On the same page as the above quote, we can consult the "Exotic Languages" table to discover that the Primordial language is spoken by Elementals and uses the Dwarvish script. Auran is the primordial dialect for the elemental plane of air, as can be gathered by the fact that air-based elementals (e.g. MM, p. 124) speak Auran.

Now, while "Aarakocra" is not "Auran", Aarakocra speak both languages. I believe that in the absence of a RAW answer, it's safe to assume that their language most likely (!) uses the same script as the Auran Primordial dialect.

Furthermore, Aarakocra does not seem to have been planned as its own language at the start of 5e - the DMG (which came out before Aarakocra were released as a player race) has guidance for creating Aarakocra NPCs, and it only gives them Auran as a default language.

Therefore, Aarakocra most likely uses the Dwarvish script.

Assuming that all scripts that exist (excluding irrelevant ones invented by individuals) are listed in the PHB on page 123, we have Common, Dwarvish, Elvish, Infernal, Celestial and Draconic.

Out of those, besides Dwarvish, Common is the only other option that makes any sense. However, considering that Aarakocra in the MM (p. 12) don't speak Common (unlike player character Aarakocras), I'd still go with the Dwarvish script.

The Aarakocra language probably has no script

By searching across all editions of D&D, the aarakocra language is mentioned only twice: in the Fiend Folio (and derivative works like the Monstrous Manual for 2nd Edition) and the Elemental Evil's Player Companion. The first source says the following about their languages:

Aarakocra speak their own language and that of the giant eagles, with which they are on mutually respectful terms. 10% of aarakocra also speak the common tongue.

The Giant Eagle language still exists in 5E, as noted in the Giant Eagle statblock. There's no mention to it having a script (that I could find).

However, most sources (across all editions) present the Aarakocra as a speakers of Auran, a dialect of Primordial – the language of Elementals – which uses the Dwarvish script (Player's Handbook, page 123). As natives of the Elemental Plane of Air, it makes sense that if they created a script for their own language, it has to be with one they are familiar with.

After all, Dwarvish script represents a strong option for the Aarakocra language.

Draconic is Likely

Based on other answers to this question, I have formed the following conclusions:

• 
  Aarakocra and Auran can't be too similar: If they were almost identical in their origin, then there would be no need for it to exist, especially since it didn't exist in previous editions and was added specifically when they became a playable race in Elemental Evil.


• 
  Auran used to use Draconic script: Finding out that Auran used to use Draconic offers itself to being an alternate script for races that derive from that plane.


• 
  Draconic script is very scratchy in style: The appearance of the Draconic script looks the way it does because dragons clawed it into stone. It makes sense that a language specific to a talon-ed race would also use Draconic script.

While there is no way to know for sure as it is not Rules as Written anywhere, I believe now that Draconic is the script of Aarakocra.