I'd like to know if playing an Aarakocra character is currently legal in the D&D 5e Adventurers League.

The Aarakocra race has been in D&D for a while, but is described in detail in 5e in the Elemental Evil Player's Companion on page 3:

This chapter presents three new races to supplement those in the Player's Handbook [...] These new options are available when you make a character, provided that your DM allows them in your campaign.