Yes, bugbear is a legal AL race
According to the D&D Adventurers League Players Guide (v 7.0) page 2:
All of the races and classes presented in the Player’s Handbook plus
one other resource from those listed below are available to build your
- Volo’s Guide to Monsters (VGM)
The only restrictions are:
Races that grant flight at 1st level, and options
presented in other resources—such as the Death
Domain found in the Dungeon Master's Guide (DMG),
or content from the various Unearthed Arcana
articles—aren’t allowed for play unless you possess
specific campaign documentation that indicates
Since bugbear is a race presented in Volo's Guide to Monsters and one that does not grant flight, it is allowed per AL rules.
Bugbear was even mentioned officially as an allowed race in
Volo's Guide to Monsters: An Official Reference for D&D Adventurers League. They retired the document when they dropped the restriction on which race could join which faction (discussed in the faction section below). As a result the document is no longer available.
Note that you are required to have access to a copy of Volo's Guide to Monsters (physically or digitally) in order to use this race. You can find the race here on dndbeyond. If you desired, you can purchase just the race from dndbeyond instead of the whole book.
Quick note about factions
I know you didn't ask, but just in case there is any confusion, in July 2018 they removed a rule in which certain monstrous races were required to join certain factions. Now, they can join any faction they want per the new “any race; any faction” mentality.